BUG: Fixed renaming of falsey names in build_table_schema (#16205)

Closes #16203

pandas/io/json/table_schema.py

@@ -76,7 +76,11 @@ def set_default_names(data):
 
 def make_field(arr, dtype=None):
     dtype = dtype or arr.dtype
-    field = {'name': arr.name or 'values',
+    if arr.name is None:
+        name = 'values'
+    else:
+        name = arr.name
+    field = {'name': name,
              'type': as_json_table_type(dtype)}
 
     if is_categorical_dtype(arr):

pandas/tests/io/json/test_json_table_schema.py

@@ -461,3 +461,11 @@ def test_overlapping_names(self):
                 data.to_json(orient='table')
 
             assert 'Overlapping' in str(excinfo.value)
+
+    def test_mi_falsey_name(self):
+        # GH 16203
+        df = pd.DataFrame(np.random.randn(4, 4),
+                          index=pd.MultiIndex.from_product([('A', 'B'),
+                                                            ('a', 'b')]))
+        result = [x['name'] for x in build_table_schema(df)['fields']]
+        assert result == ['level_0', 'level_1', 0, 1, 2, 3]