diff --git a/doc/source/whatsnew/v0.16.1.txt b/doc/source/whatsnew/v0.16.1.txt
index d00ce29fbfe92..08715e9613330 100755
--- a/doc/source/whatsnew/v0.16.1.txt
+++ b/doc/source/whatsnew/v0.16.1.txt
@@ -192,6 +192,7 @@ Bug Fixes
 - Bug in ``read_sql_table`` error when reading postgres table with timezone (:issue:`7139`)
 - Bug in ``DataFrame`` slicing may not retain metadata (:issue:`9776`)
 - Bug where ``TimdeltaIndex`` were not properly serialized in fixed ``HDFStore`` (:issue:`9635`)
+- Bug with ``TimedeltaIndex`` constructor ignoring ``name`` when given another ``TimedeltaIndex`` as data (:issue:`10025`).
 - Bug in ``DataFrameFormatter._get_formatted_index`` with not applying ``max_colwidth`` to the ``DataFrame`` index (:issue:`7856`)
 
 - Bug in ``groupby.apply()`` that would raise if a passed user defined function either returned only ``None`` (for all input). (:issue:`9685`)
diff --git a/pandas/tseries/tdi.py b/pandas/tseries/tdi.py
index 0f8ba279ec3a6..86c427682c553 100644
--- a/pandas/tseries/tdi.py
+++ b/pandas/tseries/tdi.py
@@ -140,7 +140,7 @@ def __new__(cls, data=None, unit=None,
                 copy=False, name=None,
                 closed=None, verify_integrity=True, **kwargs):
 
-        if isinstance(data, TimedeltaIndex) and freq is None:
+        if isinstance(data, TimedeltaIndex) and freq is None and name is None:
             if copy:
                 data = data.copy()
             return data
diff --git a/pandas/tseries/tests/test_timedeltas.py b/pandas/tseries/tests/test_timedeltas.py
index bc51e01ca9bdf..faf4e3fa57780 100644
--- a/pandas/tseries/tests/test_timedeltas.py
+++ b/pandas/tseries/tests/test_timedeltas.py
@@ -949,6 +949,10 @@ def test_constructor_name(self):
                             name='TEST')
         self.assertEqual(idx.name, 'TEST')
 
+        # GH10025
+        idx2 = TimedeltaIndex(idx, name='something else')
+        self.assertEqual(idx2.name, 'something else')
+
     def test_freq_conversion(self):
 
         # doc example