I would say if Kaine (or Warner) is picked as Hillary's VP and he is VP-elect, Gov. McAuliffe may appoint:
-Most likely Mark Herring.
Warner is a no-no for VP, he is the wealthiest Senator. That would not go well with Bernie supporters. His record also used to be (and still is) very moderate. He even backs TPP (but I think Kaine does too), and has a pro-trade record. Kaine also seems more charismatic, plus he speaks Spanish so that should help Clinton with Hispanic outreach (not that she needs any more with Trump topping the Republican ticket).0
He also barely survived reelection, which is even more embarrassing for someone who's long been called the most popular politician in the state and a candidate for national office.
Remember the election night? We were all like "how the hell is this happening?"