Is it solely because they hope to use it as a stepping stone to the Presidency? Because if that is the case, it seems like a plan more likely to fail than not. Only two former Vice Presidents (Nixon and the first Bush) have been elected President in the last 150 years. All of the other Vice Presidents that succeeded to the Presidency did so only because of the death or resignation of the President.
In the current election, Joe Biden has been a U.S. Senator since 1972. If Obama is elected in 2008, he will certainly run again as the incumbent in 2012. The earliest that Biden could run for President, then, is 2016, when he would be 74 years old (older than McCain in the 2008 election). Biden has a great deal of seniority in the Senate. Why would he give up such a powerful position for a chance to run for Vice President, a position famously characterized as “not worth a bucket of warm spit [piss]”?
Heck, one Vice President (Calhoun) actually resigned the position to take a seat in the Senate.
But many other politicians have given up successful political careers only to fizzle out as Vice President (the above-quoted Garner, Humphrey, Rockefeller, Mondale, Quayle, Gore, etc.).
So why do it?