I'm not sure Rand Paul was named after Ayn Rand. At any rate, unless there's real evidence it's something they'll deny (and something their opponents will bring up anyway). I'm also not sure father and son can be bracketed together as one entity with common thoughts and reactions.
It's not an easy question to answer even if we're only talking about the father: Ron Paul was and is a pretty erratic guy, and somebody who didn't bother much with details. His problem with Reagan, though, wasn't where Reagan stood or what he stood for with respect to liberals or Democrats, but that he didn't think Reagan went far enough.
In keeping with the theme here, I think you mean, DO Ron Reagan Jr. hate what his dad stood for?
>> I’m not sure Rand Paul was named after Ayn Rand <<
Me neither. I always assumed he was named for the Rand Corporation.