Use set_levels:

In [22]:
df.columns.set_levels(['b1','c1','f1'],level=1,inplace=True)
df

Out[22]:
     a         d
    b1   c1   f1
0    1    2    3
1   10   20   30
2  100  200  300

rename sets the name for the index, it doesn't rename the column names:

In [26]:
df.columns = df.columns.rename("b1", level=1)
df

Out[26]:
      a         d
b1    b    c    f
0     1    2    3
1    10   20   30
2   100  200  300

This is why you get the error

You could try:

df.columns = pd.MultiIndex.from_tuples(df.rename(columns = nested_columns).columns)
df 

Output:

  One     Two    
    a   c   b   d
0  27  67  35  36
1  80  42  93  20
2  64   9  18  83
3  85  69  60  84

Since pandas 0.21.0 the code should be like this

def rename_unnamed(df):
    """Rename unamed columns name for Pandas DataFrame

    See https://stackoverflow.com/questions/41221079/rename-multiindex-columns-in-pandas

    Parameters
    ----------
    df : pd.DataFrame object
        Input dataframe

    Returns
    -------
    pd.DataFrame
        Output dataframe

    """
    for i, columns in enumerate(df.columns.levels):
        columns_new = columns.tolist()
        for j, row in enumerate(columns_new):
            if "Unnamed: " in row:
                columns_new[j] = ""
        if pd.__version__ < "0.21.0":  # https://stackoverflow.com/a/48186976/716469
            df.columns.set_levels(columns_new, level=i, inplace=True)
        else:
            df = df.rename(columns=dict(zip(columns.tolist(), columns_new)),
                           level=i)
    return df