diff --git a/doc/source/whatsnew/v1.0.0.rst b/doc/source/whatsnew/v1.0.0.rst
index faca744a8f92c..3fe228c4e4076 100644
--- a/doc/source/whatsnew/v1.0.0.rst
+++ b/doc/source/whatsnew/v1.0.0.rst
@@ -892,7 +892,7 @@ Other
 - Fixed :class:`IntegerArray` returning ``inf`` rather than ``NaN`` for operations dividing by 0 (:issue:`27398`)
 - Fixed ``pow`` operations for :class:`IntegerArray` when the other value is ``0`` or ``1`` (:issue:`29997`)
 - Bug in :meth:`Series.count` raises if use_inf_as_na is enabled (:issue:`29478`)
-
+- Bug in :class:`Index` where a non-hashable name could be set without raising ``TypeError`` (:issue:29069`)
 
 .. _whatsnew_1000.contributors:
 
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index ce7a238daeca9..ffd3c70efe744 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -1,7 +1,7 @@
 from datetime import datetime
 import operator
 from textwrap import dedent
-from typing import FrozenSet, Union
+from typing import FrozenSet, Hashable, Optional, Union
 import warnings
 
 import numpy as np
@@ -239,7 +239,7 @@ def _outer_indexer(self, left, right):
     _typ = "index"
     _data: Union[ExtensionArray, np.ndarray]
     _id = None
-    name = None
+    _name: Optional[Hashable] = None
     _comparables = ["name"]
     _attributes = ["name"]
     _is_numeric_dtype = False
@@ -274,8 +274,7 @@ def __new__(
         from .interval import IntervalIndex
         from .category import CategoricalIndex
 
-        if name is None and hasattr(data, "name"):
-            name = data.name
+        name = maybe_extract_name(name, data, cls)
 
         if isinstance(data, ABCPandasArray):
             # ensure users don't accidentally put a PandasArray in an index.
@@ -520,7 +519,7 @@ def _simple_new(cls, values, name=None, dtype=None):
         # data buffers and strides. We don't re-use `_ndarray_values`, since
         # we actually set this value too.
         result._index_data = values
-        result.name = name
+        result._name = name
 
         return result._reset_identity()
 
@@ -1209,6 +1208,15 @@ def to_frame(self, index=True, name=None):
     # --------------------------------------------------------------------
     # Name-Centric Methods
 
+    @property
+    def name(self):
+        return self._name
+
+    @name.setter
+    def name(self, value):
+        maybe_extract_name(value, None, type(self))
+        self._name = value
+
     def _validate_names(self, name=None, names=None, deep=False):
         """
         Handles the quirks of having a singular 'name' parameter for general
@@ -1258,7 +1266,7 @@ def _set_names(self, values, level=None):
         for name in values:
             if not is_hashable(name):
                 raise TypeError(f"{type(self).__name__}.name must be a hashable type")
-        self.name = values[0]
+        self._name = values[0]
 
     names = property(fset=_set_names, fget=_get_names)
 
@@ -1546,7 +1554,7 @@ def droplevel(self, level=0):
             if mask.any():
                 result = result.putmask(mask, np.nan)
 
-            result.name = new_names[0]
+            result._name = new_names[0]
             return result
         else:
             from .multi import MultiIndex
@@ -1777,7 +1785,7 @@ def __setstate__(self, state):
                 nd_state, own_state = state
                 data = np.empty(nd_state[1], dtype=nd_state[2])
                 np.ndarray.__setstate__(data, nd_state)
-                self.name = own_state[0]
+                self._name = own_state[0]
 
             else:  # pragma: no cover
                 data = np.empty(state)
@@ -5462,3 +5470,19 @@ def default_index(n):
     from pandas.core.indexes.range import RangeIndex
 
     return RangeIndex(0, n, name=None)
+
+
+def maybe_extract_name(name, obj, cls) -> Optional[Hashable]:
+    """
+    If no name is passed, then extract it from data, validating hashability.
+    """
+    if name is None and isinstance(obj, (Index, ABCSeries)):
+        # Note we don't just check for "name" attribute since that would
+        #  pick up e.g. dtype.name
+        name = obj.name
+
+    # GH#29069
+    if not is_hashable(name):
+        raise TypeError(f"{cls.__name__}.name must be a hashable type")
+
+    return name
diff --git a/pandas/core/indexes/category.py b/pandas/core/indexes/category.py
index dd917a524e491..ba476f9e25ee6 100644
--- a/pandas/core/indexes/category.py
+++ b/pandas/core/indexes/category.py
@@ -29,7 +29,7 @@
 from pandas.core.base import _shared_docs
 import pandas.core.common as com
 import pandas.core.indexes.base as ibase
-from pandas.core.indexes.base import Index, _index_shared_docs
+from pandas.core.indexes.base import Index, _index_shared_docs, maybe_extract_name
 import pandas.core.missing as missing
 from pandas.core.ops import get_op_result_name
 
@@ -175,8 +175,7 @@ def __new__(
 
         dtype = CategoricalDtype._from_values_or_dtype(data, categories, ordered, dtype)
 
-        if name is None and hasattr(data, "name"):
-            name = data.name
+        name = maybe_extract_name(name, data, cls)
 
         if not is_categorical_dtype(data):
             # don't allow scalars
diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index 3bf6dce00a031..0e079f752edcc 100644
--- a/pandas/core/indexes/datetimelike.py
+++ b/pandas/core/indexes/datetimelike.py
@@ -814,7 +814,6 @@ class DatetimelikeDelegateMixin(PandasDelegate):
     _raw_methods: Set[str] = set()
     # raw_properties : dispatch properties that shouldn't be boxed in an Index
     _raw_properties: Set[str] = set()
-    name = None
     _data: ExtensionArray
 
     @property
diff --git a/pandas/core/indexes/datetimes.py b/pandas/core/indexes/datetimes.py
index 1fd962dd24656..d226cf7d940f1 100644
--- a/pandas/core/indexes/datetimes.py
+++ b/pandas/core/indexes/datetimes.py
@@ -30,7 +30,7 @@
 )
 from pandas.core.base import _shared_docs
 import pandas.core.common as com
-from pandas.core.indexes.base import Index
+from pandas.core.indexes.base import Index, maybe_extract_name
 from pandas.core.indexes.datetimelike import (
     DatetimeIndexOpsMixin,
     DatetimelikeDelegateMixin,
@@ -254,8 +254,7 @@ def __new__(
 
         # - Cases checked above all return/raise before reaching here - #
 
-        if name is None and hasattr(data, "name"):
-            name = data.name
+        name = maybe_extract_name(name, data, cls)
 
         dtarr = DatetimeArray._from_sequence(
             data,
diff --git a/pandas/core/indexes/interval.py b/pandas/core/indexes/interval.py
index b61e80b9e89a7..221ad564a56f4 100644
--- a/pandas/core/indexes/interval.py
+++ b/pandas/core/indexes/interval.py
@@ -47,6 +47,7 @@
     _index_shared_docs,
     default_pprint,
     ensure_index,
+    maybe_extract_name,
 )
 from pandas.core.indexes.datetimes import DatetimeIndex, date_range
 from pandas.core.indexes.multi import MultiIndex
@@ -217,8 +218,7 @@ def __new__(
         verify_integrity: bool = True,
     ):
 
-        if name is None and hasattr(data, "name"):
-            name = data.name
+        name = maybe_extract_name(name, data, cls)
 
         with rewrite_exception("IntervalArray", cls.__name__):
             array = IntervalArray(
diff --git a/pandas/core/indexes/numeric.py b/pandas/core/indexes/numeric.py
index 00d81f3ed95a9..39cbe5f151262 100644
--- a/pandas/core/indexes/numeric.py
+++ b/pandas/core/indexes/numeric.py
@@ -30,7 +30,12 @@
 
 from pandas.core import algorithms
 import pandas.core.common as com
-from pandas.core.indexes.base import Index, InvalidIndexError, _index_shared_docs
+from pandas.core.indexes.base import (
+    Index,
+    InvalidIndexError,
+    _index_shared_docs,
+    maybe_extract_name,
+)
 from pandas.core.ops import get_op_result_name
 
 _num_index_shared_docs = dict()
@@ -68,8 +73,7 @@ def __new__(cls, data=None, dtype=None, copy=False, name=None):
         else:
             subarr = data
 
-        if name is None and hasattr(data, "name"):
-            name = data.name
+        name = maybe_extract_name(name, data, cls)
         return cls._simple_new(subarr, name=name)
 
     @classmethod
diff --git a/pandas/core/indexes/period.py b/pandas/core/indexes/period.py
index 9485116a8084a..6465a0c1724af 100644
--- a/pandas/core/indexes/period.py
+++ b/pandas/core/indexes/period.py
@@ -25,7 +25,11 @@
 from pandas.core.base import _shared_docs
 import pandas.core.common as com
 import pandas.core.indexes.base as ibase
-from pandas.core.indexes.base import _index_shared_docs, ensure_index
+from pandas.core.indexes.base import (
+    _index_shared_docs,
+    ensure_index,
+    maybe_extract_name,
+)
 from pandas.core.indexes.datetimelike import (
     DatetimeIndexOpsMixin,
     DatetimelikeDelegateMixin,
@@ -184,8 +188,7 @@ def __new__(
             argument = list(set(fields) - valid_field_set)[0]
             raise TypeError(f"__new__() got an unexpected keyword argument {argument}")
 
-        if name is None and hasattr(data, "name"):
-            name = data.name
+        name = maybe_extract_name(name, data, cls)
 
         if data is None and ordinal is None:
             # range-based.
diff --git a/pandas/core/indexes/range.py b/pandas/core/indexes/range.py
index 6ad70841a48b0..225cd43bbd588 100644
--- a/pandas/core/indexes/range.py
+++ b/pandas/core/indexes/range.py
@@ -26,7 +26,7 @@
 import pandas.core.common as com
 from pandas.core.construction import extract_array
 import pandas.core.indexes.base as ibase
-from pandas.core.indexes.base import Index, _index_shared_docs
+from pandas.core.indexes.base import Index, _index_shared_docs, maybe_extract_name
 from pandas.core.indexes.numeric import Int64Index
 from pandas.core.ops.common import unpack_zerodim_and_defer
 
@@ -85,10 +85,10 @@ def __new__(
     ):
 
         cls._validate_dtype(dtype)
+        name = maybe_extract_name(name, start, cls)
 
         # RangeIndex
         if isinstance(start, RangeIndex):
-            name = start.name if name is None else name
             start = start._range
             return cls._simple_new(start, dtype=dtype, name=name)
 
diff --git a/pandas/core/indexes/timedeltas.py b/pandas/core/indexes/timedeltas.py
index 889075ebe4e31..2569a510fdffb 100644
--- a/pandas/core/indexes/timedeltas.py
+++ b/pandas/core/indexes/timedeltas.py
@@ -25,7 +25,7 @@
 from pandas.core.arrays.timedeltas import TimedeltaArray, _is_convertible_to_td
 from pandas.core.base import _shared_docs
 import pandas.core.common as com
-from pandas.core.indexes.base import Index, _index_shared_docs
+from pandas.core.indexes.base import Index, _index_shared_docs, maybe_extract_name
 from pandas.core.indexes.datetimelike import (
     DatetimeIndexOpsMixin,
     DatetimelikeDelegateMixin,
@@ -163,6 +163,7 @@ def __new__(
         copy=False,
         name=None,
     ):
+        name = maybe_extract_name(name, data, cls)
 
         if is_scalar(data):
             raise TypeError(
@@ -210,7 +211,7 @@ def _simple_new(cls, values, name=None, freq=None, dtype=_TD_DTYPE):
         tdarr = TimedeltaArray._simple_new(values._data, freq=freq)
         result = object.__new__(cls)
         result._data = tdarr
-        result.name = name
+        result._name = name
         # For groupby perf. See note in indexes/base about _index_data
         result._index_data = tdarr._data
 
diff --git a/pandas/core/series.py b/pandas/core/series.py
index 54c163330e6ee..a8aab5f216458 100644
--- a/pandas/core/series.py
+++ b/pandas/core/series.py
@@ -194,6 +194,8 @@ def __init__(
 
         else:
 
+            name = ibase.maybe_extract_name(name, data, type(self))
+
             if is_empty_data(data) and dtype is None:
                 # gh-17261
                 warnings.warn(
@@ -219,8 +221,6 @@ def __init__(
                     "initializing a Series from a MultiIndex is not supported"
                 )
             elif isinstance(data, Index):
-                if name is None:
-                    name = data.name
 
                 if dtype is not None:
                     # astype copies
@@ -237,8 +237,6 @@ def __init__(
             elif isinstance(data, np.ndarray):
                 pass
             elif isinstance(data, ABCSeries):
-                if name is None:
-                    name = data.name
                 if index is None:
                     index = data.index
                 else:
diff --git a/pandas/tests/indexes/common.py b/pandas/tests/indexes/common.py
index 102949fe3f05e..68cca473d6bb0 100644
--- a/pandas/tests/indexes/common.py
+++ b/pandas/tests/indexes/common.py
@@ -103,6 +103,13 @@ def test_shift(self):
         with pytest.raises(NotImplementedError, match=msg):
             idx.shift(1, 2)
 
+    def test_constructor_name_unhashable(self):
+        # GH#29069 check that name is hashable
+        # See also same-named test in tests.series.test_constructors
+        idx = self.create_index()
+        with pytest.raises(TypeError, match="Index.name must be a hashable type"):
+            type(idx)(idx, name=[])
+
     def test_create_index_existing_name(self):
 
         # GH11193, when an existing index is passed, and a new name is not
diff --git a/pandas/tests/indexes/test_base.py b/pandas/tests/indexes/test_base.py
index d4d644e486478..6ec35a32d74ce 100644
--- a/pandas/tests/indexes/test_base.py
+++ b/pandas/tests/indexes/test_base.py
@@ -100,6 +100,7 @@ def test_constructor_copy(self, index):
         arr[0] = "SOMEBIGLONGSTRING"
         assert new_index[0] != "SOMEBIGLONGSTRING"
 
+        # FIXME: dont leave commented-out
         # what to do here?
         # arr = np.array(5.)
         # pytest.raises(Exception, arr.view, Index)