Dan Quayle IOTL came under some criticism for his lack of experience (the infamous "you're no Jack Kennedy" remark demonstrates the effectiveness of his attempt to spin that in his favor), and as Vice President he would act in a manner that led many to question his actual intelligence.
Suppose he says something phenomenally stupid to George H. W. Bush while Bush is looking for running mates in 1988, dumb enough to get Bush the Elder to write him off.
Who else can Bush Sr. choose as his running mate, and would that running mate help or hurt him in 1988 and 1992?