Don't get me wrong, the role of Vice President is a vital role for our country, one that is only a hearbeat away to the highest office in the land for that individual. I say it really does not matter because we are talking about Donald Trump here. He has said on numerous occassions that he will go with his gut. Doesn't sound like he is willing to take any advice, so other then being a 'high profile dignitary' for dedicating monuments and such, what else would that person really be doing?
Donald Trump should choose Indiana's Governor, Mike Pence, as his running mate. Trump needs to pick someone that will make his ticket more serious. This pick would show voters that a Trump Administration could handle policy decisions. Furthermore, Pence has solid connections between the Republican Party's establishment and the Tea Party members. Therefore, it would be wide for Donald Trump to select Mike Pence to be his running mate.
Mike Pence will be a good rational choice for Donald Trump to have as a running mate. Pence is a seasoned politician and can offer some balance to the platform. Trump needs a running mate that is well connected and knows his way around the political scene of DC. The GOP will be more receptive of Trump if Mike Pence is the VP.
Trump is doing quite well with white males, but because of his apparent racism and sexism, he is struggling with minorities and women. Picking another white male as his running mate probably won't help him with those groups. Now, picking a woman or ethnic minority may not earn him a whole lot of votes, but I think it would certainly help him out more than an uninteresting pick like Pence.