diff --git a/doc/source/whatsnew/v0.25.0.rst b/doc/source/whatsnew/v0.25.0.rst
index ea08a0a6fe07b..82bfe526cf161 100644
--- a/doc/source/whatsnew/v0.25.0.rst
+++ b/doc/source/whatsnew/v0.25.0.rst
@@ -86,7 +86,7 @@ Other API Changes
 - :class:`DatetimeTZDtype` will now standardize pytz timezones to a common timezone instance (:issue:`24713`)
 - ``Timestamp`` and ``Timedelta`` scalars now implement the :meth:`to_numpy` method as aliases to :meth:`Timestamp.to_datetime64` and :meth:`Timedelta.to_timedelta64`, respectively. (:issue:`24653`)
 - :meth:`Timestamp.strptime` will now rise a ``NotImplementedError`` (:issue:`25016`)
--
+- Bug in :meth:`DatetimeIndex.snap` which didn't preserving the ``name`` of the input :class:`Index` (:issue:`25575`)
 
 .. _whatsnew_0250.deprecations:
 
diff --git a/pandas/core/indexes/datetimes.py b/pandas/core/indexes/datetimes.py
index b8d052ce7be04..b65e59a3d58b7 100644
--- a/pandas/core/indexes/datetimes.py
+++ b/pandas/core/indexes/datetimes.py
@@ -787,8 +787,8 @@ def snap(self, freq='S'):
             snapped[i] = s
 
         # we know it conforms; skip check
-        return DatetimeIndex._simple_new(snapped, freq=freq)
-        # TODO: what about self.name?  tz? if so, use shallow_copy?
+        return DatetimeIndex._simple_new(snapped, name=self.name, tz=self.tz,
+                                         freq=freq)
 
     def join(self, other, how='left', level=None, return_indexers=False,
              sort=False):
diff --git a/pandas/tests/series/indexing/test_datetime.py b/pandas/tests/series/indexing/test_datetime.py
index 0efc9feb0dbd4..8e4c7d9b17efc 100644
--- a/pandas/tests/series/indexing/test_datetime.py
+++ b/pandas/tests/series/indexing/test_datetime.py
@@ -52,20 +52,28 @@ def test_fancy_setitem():
     assert (s[48:54] == -3).all()
 
 
-def test_dti_snap():
+@pytest.mark.filterwarnings("ignore::DeprecationWarning")
+@pytest.mark.parametrize('tz', [None, 'Asia/Shanghai', 'Europe/Berlin'])
+@pytest.mark.parametrize('name', [None, 'my_dti'])
+def test_dti_snap(name, tz):
     dti = DatetimeIndex(['1/1/2002', '1/2/2002', '1/3/2002', '1/4/2002',
-                         '1/5/2002', '1/6/2002', '1/7/2002'], freq='D')
+                         '1/5/2002', '1/6/2002', '1/7/2002'],
+                        name=name, tz=tz, freq='D')
 
-    res = dti.snap(freq='W-MON')
-    exp = date_range('12/31/2001', '1/7/2002', freq='w-mon')
-    exp = exp.repeat([3, 4])
-    assert (res == exp).all()
+    result = dti.snap(freq='W-MON')
+    expected = date_range('12/31/2001', '1/7/2002',
+                          name=name, tz=tz, freq='w-mon')
+    expected = expected.repeat([3, 4])
+    tm.assert_index_equal(result, expected)
+    assert result.tz == expected.tz
 
-    res = dti.snap(freq='B')
+    result = dti.snap(freq='B')
 
-    exp = date_range('1/1/2002', '1/7/2002', freq='b')
-    exp = exp.repeat([1, 1, 1, 2, 2])
-    assert (res == exp).all()
+    expected = date_range('1/1/2002', '1/7/2002',
+                          name=name, tz=tz, freq='b')
+    expected = expected.repeat([1, 1, 1, 2, 2])
+    tm.assert_index_equal(result, expected)
+    assert result.tz == expected.tz
 
 
 def test_dti_reset_index_round_trip():