There was an interesting hypothetical on a political forum: What if nominees for Vice President were selected in primaries the way many states select candidates for Lieutenant Governor?
It highlights the problems with the method many states have of selecting Lieutenant Governors. Those should be replaced by a system where a candidate for Governor chooses a running mate after securing the nomination, so that the pool of potential nominees can include people who lost primaries for other prestigious posts, rather than limiting to those who initially see Lt. Governor as their best shot. It could very well be that an also-ran for Governor, Senate, US House, or Attorney General has greater political talent than those who pick this one particular office.
There is a bit of a distinction that Lieutenant Governors have specific responsibilities, like presiding over the State Senate, whereas a Vice President’s power can be determined by the strength of their relationship with the president. So it may make more sense to have these primaries, although this rarely comes up in the campaigns.
There are additional issues with applying a primary system to Vice Presidents, which would prevent some recent nominees from being selected. The VP would be someone who has been campaigning for the post for at least an year before the election, which excludes presidential primary also-rans (Ronald Reagan picked George HW Bush, John Kerry picked John Edwards, Barack Obama picked Joe Biden), retired statesmen who might be talked into campaigning for a few months (Jack Kemp, Dick Cheney) but not longer, and statewide officeholders/ prominent cabinet-congressional members who might be uncomfortable spending over an year running for non-presidential national office, or for promising to serve with any presidential candidate their party selects. Mike Pence and Tim Kaine probably would have been willing to do it, though, so it might not have changed too much in 2016.
The constitutional restriction against electors voting for people from the same state also complicates matters. How would it be coordinated that two California Democrats or two New York Democrats or two Texas Republicans don’t win both spots on the ticket, resulting in a potentially difficult situation with electors who are legally not allowed to vote for candidates for President and Vice President from their state?
One further problem is that the VP candidate might end up being a poor match for the top of the ticket. In 2012, John Huntsman could have been a good fit for Rick Santorum, Newt Gingrich or Tim Pawlenty, but not Mitt Romney, a fellow Mormon businessman. There could be weirdness if the same ethnic or religious minorities were on both the top and bottom of the ticket, without either candidate wanting that outcome. You might also have two candidates for different offices trying to sabotage one another during the primary process, which isn’t conducive to party unity.