diff --git a/pandas/core/frame.py b/pandas/core/frame.py
index 76f68fdaa7845..803d1c914c954 100644
--- a/pandas/core/frame.py
+++ b/pandas/core/frame.py
@@ -5073,10 +5073,6 @@ def drop(
             errors=errors,
         )
 
-    @rewrite_axis_style_signature(
-        "mapper",
-        [("copy", True), ("inplace", False), ("level", None), ("errors", "ignore")],
-    )
     def rename(
         self,
         mapper: Renamer | None = None,
diff --git a/pandas/core/groupby/groupby.py b/pandas/core/groupby/groupby.py
index faa32b31a73d7..1dafe1618a9f8 100644
--- a/pandas/core/groupby/groupby.py
+++ b/pandas/core/groupby/groupby.py
@@ -2082,7 +2082,8 @@ def size(self) -> DataFrame | Series:
             result = self._obj_1d_constructor(result)
 
         if not self.as_index:
-            result = result.rename("size").reset_index()
+            # Item "None" of "Optional[Series]" has no attribute "reset_index"
+            result = result.rename("size").reset_index()  # type: ignore[union-attr]
 
         return self._reindex_output(result, fill_value=0)
 
diff --git a/pandas/core/series.py b/pandas/core/series.py
index 03fac7cceabb7..d5909b8659903 100644
--- a/pandas/core/series.py
+++ b/pandas/core/series.py
@@ -4468,14 +4468,16 @@ def align(
 
     def rename(
         self,
-        index=None,
+        mapper=None,
         *,
+        index=None,
+        columns=None,
         axis=None,
         copy=True,
         inplace=False,
         level=None,
         errors="ignore",
-    ):
+    ) -> Series | None:
         """
         Alter Series index labels or name.
 
@@ -4491,7 +4493,7 @@ def rename(
         ----------
         axis : {0 or "index"}
             Unused. Accepted for compatibility with DataFrame method only.
-        index : scalar, hashable sequence, dict-like or function, optional
+        mapper : scalar, hashable sequence, dict-like or function, optional
             Functions or dict-like are transformations to apply to
             the index.
             Scalar or hashable sequence-like will alter the ``Series.name``
@@ -4539,12 +4541,16 @@ def rename(
             # Make sure we raise if an invalid 'axis' is passed.
             axis = self._get_axis_number(axis)
 
-        if callable(index) or is_dict_like(index):
+        if index is not None and mapper is not None:
+            raise TypeError("Cannot specify both 'mapper' and 'index'")
+        if mapper is None:
+            mapper = index
+        if callable(mapper) or is_dict_like(mapper):
             return super().rename(
-                index, copy=copy, inplace=inplace, level=level, errors=errors
+                mapper, copy=copy, inplace=inplace, level=level, errors=errors
             )
         else:
-            return self._set_name(index, inplace=inplace)
+            return self._set_name(mapper, inplace=inplace)
 
     @overload
     def set_axis(
diff --git a/pandas/io/json/_normalize.py b/pandas/io/json/_normalize.py
index 2c2c127394fb6..36a7949a9f1e3 100644
--- a/pandas/io/json/_normalize.py
+++ b/pandas/io/json/_normalize.py
@@ -517,7 +517,11 @@ def _recursive_extract(data, path, seen_meta, level=0):
     result = DataFrame(records)
 
     if record_prefix is not None:
-        result = result.rename(columns=lambda x: f"{record_prefix}{x}")
+        # Incompatible types in assignment (expression has type "Optional[DataFrame]",
+        # variable has type "DataFrame")
+        result = result.rename(  # type: ignore[assignment]
+            columns=lambda x: f"{record_prefix}{x}"
+        )
 
     # Data types, a problem
     for k, v in meta_vals.items():
diff --git a/pandas/tests/series/methods/test_rename.py b/pandas/tests/series/methods/test_rename.py
index a78abfa63cff4..3425dd8f019e7 100644
--- a/pandas/tests/series/methods/test_rename.py
+++ b/pandas/tests/series/methods/test_rename.py
@@ -105,6 +105,19 @@ def test_rename_callable(self):
 
         assert result.name == expected.name
 
+    def test_rename_method_and_index(self):
+        # GH 40977
+        ser = Series([1, 2])
+        with pytest.raises(TypeError, match="Cannot specify both 'mapper' and 'index'"):
+            ser.rename(str, index=str)
+
+    def test_rename_none(self):
+        # GH 40977
+        ser = Series([1, 2], name="foo")
+        result = ser.rename(None)
+        expected = Series([1, 2])
+        tm.assert_series_equal(result, expected)
+
     def test_rename_series_with_multiindex(self):
         # issue #43659
         arrays = [