If Obama’s father were white, he no more would have been considered Presidential timber than if the last President’s father had been named Smith.
Surely, you meant "Presidential timbre."
I tried both phrases in a search engine, and found about 20 times more examples of "Presidential timber" than "Presidential timbre." Here's a debate between an English teacher and a Reuter's copy editor over the question.
My offhand guess would be that the phrase started out as "Presidential timbre," but all the lumber-related Presidential folklore -- Washington chopping down the cherry tree, Lincoln splitting rails -- led to a change in spelling over the years.
On the other hand, almost all cute etymological theories like this turn out to be wrong.