There is no doubt that, on the whole, recent Vice-Presidents have played a greater role in American governance than before. Al Gore used to claim to be the most involved Vice-President in history. That was not strictly true even at the time he made the claim - he was second to George H W Bush - and is plainly ridiculous now, but it does show a trend.
Three of the last four Vice-Presidents (Quayle, of course, is the exception) have held very significant positions of power. If this becomes the expectation, then people's view of the position will change, and this could have electoral consequences.
For example, it may become more common for presidential candidates to declare a running mate early in the primary season. This has one clear disadvantage. The only people available early in the campaign are people who already support your campaign. You could not - as Kerry did in 2004 and Reagan in 1980 - have your rival as your running mate. But if there is an expectation that the Vice-President will have a major role in the government then a choice of running mate can be used to address a candidate's weakness within the party, as well as with the general electorate.
For example, George W Bush's biggest areas of weakness in 2000 were lack of Washington, and especially national security, expertise. He chose as his running mate a former Congressman, WH Chief of Staff, and Defense Secretary. His father had been chosen by Ronald Reagan on a similar basis.
If you look at some of the leading contenders for the Republican nomination, it is easy to see how they could address their weaknesses.
Rudi Giuliani - not trusted by conservatives, especially in the South; no DC experience. Solution: Giuliani-Gingrich; Giuliani-Pence.
John McCain - not trusted by conservatives, especially in the South; not trusted by party activists; no state level experience. Solution: McCain-Barbour; McCain-Bush (JEB, obviously).
Mitt Romney - conservative jury still out on him; no Washington or national security experience. Romney-Rice; Romney-Cheney.
Newt Gingrich - deeply polarising figure with little appeal to cross-over voters; no state level experience. Solution: Gingrich-Giuliani; Gingrich-Romney.