As bizarre soul who actually blew some of his vacation in '88 on a Florida beach that summer watching the Rep convention coverage..
VP choices are generally all about compensating for the POTUS candidate's perceived weaknesses. Bush had been a moderate before his association with Reagan, which meant the conservative base of the Republican party was not all that hyped over him. So it was pretty much a forgone conclusion that his VP candidate had to be a true Conservative. Quayle certainly was (and is) that. So he was a good choice to shore up the base, but pretty much any conservative would have been just as good there.
But Bush was at this time 64 years old, and had just come off of an 8-year stint as VP, which is guaranteed to make just about anybody look dull (Biden somehow magically excepted). The Doonesbury caricature of him was nothing but a disembodied voice. Also, the avidly anti-abortion and bellicose Reagan administration had opened up a large gender-gap between the parties. So you can see where it may have been thought the ticket needed some pizazz, and a way to appeal to women.
With staged political events like the party nominating conventions, when information is initially released by the party elders, it comes complete with some points of discussion. This way they can at least get the media talking about what they want talked about from the start. Thus you can generally get some clues to what the decision makers thought the best selling-points for an idea (eg: a VP choice) would have been by what the initial discussions about the idea are.
In this case, right from the beginning, even in some of the network coverage of the leaks before the official announcement, all the talk was about how Quayle was young and telegenic, and "resembled JFK", and how this would appeal to female voters. So that is most likely what they were thinking. (One wonders how many, if any, of the people making this supposition in the RNC's smoke-filled rooms in '88 were actual females).
There were some superficial resemblances. In particular, at the time JFK ran for president he was roughly the same age as Quayle, and had served in the senate roughly the same amount of time. They were both white males too. Unfortunately for the GHWB brain-trust, that's pretty much where the resemblance ended.
However, the JFK comparisons stayed in Quayle's stump speeches (VP stump speeches generally have to be approved by the POTUS candidate's people, when they aren't just flat out written by them). He even made the comparison during his VP debate performance.
This of course set the stage for the top classic moment in all of US Presidential Debate history, Loyd Bentzen's "Senator, you're no Jack Kennedy" line. (youtube)