diff --git a/seaborn/_core/data.py b/seaborn/_core/data.py
index 6c1cf87217..6fe0a0a5b5 100644
--- a/seaborn/_core/data.py
+++ b/seaborn/_core/data.py
@@ -132,10 +132,11 @@ def _assign_variables(
         names: dict[str, str | None]
 
         plot_data = {}
-        var_names = {}
+        names = {}
 
         # Data is optional; all variables can be defined as vectors
-        if data is None:
+        data_not_passed = data is None
+        if data_not_passed:
             data = {}
 
         # TODO Generally interested in accepting a generic DataFrame interface
@@ -172,23 +173,28 @@ def _assign_variables(
                     plot_data[key] = data[val]
                 elif val in index:
                     plot_data[key] = index[val]
-                var_names[key] = str(val)
+                names[key] = str(val)
 
             elif isinstance(val, str):
 
-                # This looks like a column name but we don't know what it means!
-                # TODO improve this feedback to distinguish between
-                # - "you passed a string, but did not pass data"
-                # - "you passed a string, it was not found in data"
+                # This looks like a column name but, lookup failed.
 
-                err = f"Could not interpret value `{val}` for parameter `{key}`"
+                err = f"Could not interpret value `{val}` for `{key}`. "
+                if data_not_passed:
+                    err += "Value is a string, but `data` was not passed."
+                else:
+                    err += "An entry with this name does not appear in `data`."
                 raise ValueError(err)
 
             else:
 
-                # Otherwise, assume the value is itself data
+                # Otherwise, assume the value somehow represents data
+
+                # Ignore empty data structures
+                if isinstance(val, abc.Sized) and len(val) == 0:
+                    continue
 
-                # Raise when data object is present and a vector can't matched
+                # If vector has no index, it must match length of data table
                 if isinstance(data, pd.DataFrame) and not isinstance(val, pd.Series):
                     if isinstance(val, abc.Sized) and len(data) != len(val):
                         val_cls = val.__class__.__name__
@@ -201,19 +207,14 @@ def _assign_variables(
 
                 plot_data[key] = val
 
-                # Try to infer the name of the variable
-                var_names[key] = getattr(val, "name", None)
+                # Try to infer the original name using pandas-like metadata
+                if hasattr(val, "name"):
+                    names[key] = str(val.name)
+                else:
+                    names[key] = None
 
         # Construct a tidy plot DataFrame. This will convert a number of
         # types automatically, aligning on index in case of pandas objects
         frame = pd.DataFrame(plot_data)
 
-        # Build the mapping from plot variables to original names (if available)
-        names = {
-            var: name
-            for var, name in var_names.items()
-            # TODO I am not sure that this is necessary any more
-            if frame[var].notnull().any()
-        }
-
         return frame, names
diff --git a/seaborn/tests/_core/test_data.py b/seaborn/tests/_core/test_data.py
index 7ba4c0de54..4c89a16924 100644
--- a/seaborn/tests/_core/test_data.py
+++ b/seaborn/tests/_core/test_data.py
@@ -190,7 +190,7 @@ def test_key_not_in_data_raises(self, long_df):
 
         var = "x"
         key = "what"
-        msg = f"Could not interpret value `{key}` for parameter `{var}`"
+        msg = f"Could not interpret value `{key}` for `{var}`. An entry with this name"
         with pytest.raises(ValueError, match=msg):
             PlotData(long_df, {var: key})
 
@@ -198,9 +198,9 @@ def test_key_with_no_data_raises(self):
 
         var = "x"
         key = "what"
-        msg = f"Could not interpret value `{key}` for parameter `{var}`"
+        msg = f"Could not interpret value `{key}` for `{var}`. Value is a string,"
         with pytest.raises(ValueError, match=msg):
-            PlotData(variables={var: key})
+            PlotData(None, {var: key})
 
     def test_data_vector_different_lengths_raises(self, long_df):