Google does not answer everything. Consider:
"Wilson" 204.000.000
"George W. Bush" 47.500.000 - "Bush" 302.000.000
"Winston Churchill" 9.650.000 - "Churchill" 34.600.000
So what is the more common name?
Not counting in each case that of course any hit for "Thomas Woodrow Wilson" will be included in "Woodrow Wilson" too.
I can respect disagreement but not silly and lazy pseudo-arguments like a google search. 
Since Grover Cleveland has been used as an example: Grover was one of his first names, whereas Woodrow is Wilson's mother's maiden name. (talk)