What about a prominent (former) White House Chief of Staff?
Let's say, instead of running for mayor of Chicago, Rahm Emanuel stayed on as CoS for Obama's entire first term, and then he started to prepare for his own presidential run following the 2012 election.
A la Leo McGarry in West Wing. Personally I thought that was a little far-fetched considering his personality and the position's relatively low profile considering it's ultimately a behind-the-scenes role.
I don't know. Leo being on the ticket did make a certain amount a sense to me.
First, he was only the VP candidate and not running for president himself. Second, he seemed to be one of the more influential, "prime ministerial" Chiefs of Staffs. And third, he did have a political career of his own prior to becoming Bartlet's Chief of Staff.
He was a former Secretary of Labor who talked Bartlet into running for president, managed his campaign and then became his CoS. It was never canonically established what Leo did before his tenure as Secretary of Labor, but of course you're not appointed to the Cabinet out of the blue. So, it's likely that he held some other office prior to that too.
He probably wasn't extremely well-known in the general public, but then again that's not necessarily the case with VP candidates either. How big was Paul Ryan's name recognition on the national level prior to 2012?
