diff --git a/doc/source/whatsnew/v2.0.2.rst b/doc/source/whatsnew/v2.0.2.rst
index e791e2a58ba5b..2b07622d70c3f 100644
--- a/doc/source/whatsnew/v2.0.2.rst
+++ b/doc/source/whatsnew/v2.0.2.rst
@@ -30,6 +30,7 @@ Bug fixes
 - Bug in :func:`api.interchange.from_dataframe` was unnecessarily raising on bitmasks (:issue:`49888`)
 - Bug in :meth:`DataFrame.convert_dtypes` ignores ``convert_*`` keywords when set to False ``dtype_backend="pyarrow"`` (:issue:`52872`)
 - Bug in :meth:`Series.describe` treating pyarrow-backed timestamps and timedeltas as categorical data (:issue:`53001`)
+- Bug in :meth:`Series.rename` not making a lazy copy when Copy-on-Write is enabled when a scalar is passed to it (:issue:`52450`)
 - Bug in :meth:`pd.array` raising for ``NumPy`` array and ``pa.large_string`` or ``pa.large_binary`` (:issue:`52590`)
 - Bug in :meth:`DataFrame.__getitem__` not preserving dtypes for :class:`MultiIndex` partial keys (:issue:`51895`)
 -
diff --git a/pandas/core/series.py b/pandas/core/series.py
index 29e02fdc7695d..78f4da4e65196 100644
--- a/pandas/core/series.py
+++ b/pandas/core/series.py
@@ -1940,7 +1940,9 @@ def to_frame(self, name: Hashable = lib.no_default) -> DataFrame:
         df = self._constructor_expanddim(mgr)
         return df.__finalize__(self, method="to_frame")
 
-    def _set_name(self, name, inplace: bool = False) -> Series:
+    def _set_name(
+        self, name, inplace: bool = False, deep: bool | None = None
+    ) -> Series:
         """
         Set the Series name.
 
@@ -1949,9 +1951,11 @@ def _set_name(self, name, inplace: bool = False) -> Series:
         name : str
         inplace : bool
             Whether to modify `self` directly or return a copy.
+        deep : bool|None, default None
+            Whether to do a deep copy, a shallow copy, or Copy on Write(None)
         """
         inplace = validate_bool_kwarg(inplace, "inplace")
-        ser = self if inplace else self.copy()
+        ser = self if inplace else self.copy(deep and not using_copy_on_write())
         ser.name = name
         return ser
 
@@ -4770,7 +4774,7 @@ def rename(
         index: Renamer | Hashable | None = None,
         *,
         axis: Axis | None = None,
-        copy: bool = True,
+        copy: bool | None = None,
         inplace: bool = False,
         level: Level | None = None,
         errors: IgnoreRaise = "ignore",
@@ -4857,7 +4861,7 @@ def rename(
                 errors=errors,
             )
         else:
-            return self._set_name(index, inplace=inplace)
+            return self._set_name(index, inplace=inplace, deep=copy)
 
     @Appender(
         """
diff --git a/pandas/tests/copy_view/test_methods.py b/pandas/tests/copy_view/test_methods.py
index 1d8f1dea7d478..c220c46bdc8f8 100644
--- a/pandas/tests/copy_view/test_methods.py
+++ b/pandas/tests/copy_view/test_methods.py
@@ -135,6 +135,7 @@ def test_methods_copy_keyword(
     "method",
     [
         lambda ser, copy: ser.rename(index={0: 100}, copy=copy),
+        lambda ser, copy: ser.rename(None, copy=copy),
         lambda ser, copy: ser.reindex(index=ser.index, copy=copy),
         lambda ser, copy: ser.reindex_like(ser, copy=copy),
         lambda ser, copy: ser.align(ser, copy=copy)[0],
@@ -152,6 +153,7 @@ def test_methods_copy_keyword(
         lambda ser, copy: ser.set_flags(allows_duplicate_labels=False, copy=copy),
     ],
     ids=[
+        "rename (dict)",
         "rename",
         "reindex",
         "reindex_like",