Yes, the only way I could see President Quayle is if Bush Sr. died in office. Even then, I'm not sure he'd win the election. I mean, the halo effect is one thing, but if he is as gaffe prone in the aftermath of Bush's death as he was as Vice President, he probably isn't going to win.
I remember there was a joke on weekend update in either 1999 or 2000 where they mentioned it was Al Gore's birthday. Colin Quinn said something like, "It's your birthday Mr. Gore-what did ya wish for?" And that comment was followed by a fake newspaper which announced "
Republicans Nominate Quayle"
From what I've heard from people who've worked with him, Quayle isn't as stupid in person as he appeared to be while Vice President, apparently he's more gaffe prone than outright idiotic. So if Bush died and Quayle was President between say, 1992-January 20th 1993 I don't think he'd be a total disaster.
There's no way he can win 2008. With Bush's unpopularity, I'm not sure any Republican could win.
Health problems ruled out 1996. So the only chance Quayle has is 2000. But you'd need to pull some serious strings to get Quayle the nomination. You probably have to keep W. out of the race entirely. Maybe the POD is him becoming Baseball Commissioner in 1992 and just assuming the Clinton years go more or less the same? With W. out of the race, I'm not sure who the heir apparent is. Dole obviously won't run again, and it isn't McCain. If Quayle plays his cards right he could get the kind conservative support W. got, and therefore somehow get the nomination. Honestly a McCain nomination is a hell of a lot more likely but I'm trying to think of a circumstance in which Quayle wins the nomination. Then let's say Quayle's used the time out of the Vice Presidency to really prepare for the campaign. And I mean really really prepare, so it isn't an easy win for Gore. Let's say for whatever reason Quayle wins the election.
Granted the circumstances described above are almost ludicrously unlikely. IOTL Quayle polled eighth when he ran for the nomination. And much of it depends on Quayle absorbing the "next in line"/"Conservative" support that W. received in the primaries IOTL. I'm actually think both McCain and Gore are both more likely to win in a Bushless 2000, but there are upsets and unforseen circumstances. Quayle 2000 is really really really really really really really really unlikely. But Quayle 2008 is essentially impossible.