Rand Paul can't do any better than his father.
Ron Paul couldn't win a single state. And Rand is less charmismatic, more stern and angry, less experienced, and just a worst candidate.
Except he has one thing going for him that his father didn't: His father was stuck in the "libertarian ghetto" of the party, and didn't make much effort, if any, to appeal to "regular Republicans". Rand is trying to hold on to his father's base, and expand out from that.
Regarding where he would do best, I still think the bulk of his support would likely come from the same pool of voters that his father won, and the fact that he represents Kentucky wouldn't really give him any special advantage in the South. His father tended to have his best showings in caucus states (though also did very well in the NH and VT primaries), and did especially well in the West. He didn't do so well in the South.
Here are Ron Paul's %ages from the first five states in 2012:
Iowa caucus: 21%
New Hampshire primary: 23%
South Carolina primary: 13%
Florida primary: 7%
Nevada caucus: 19%
In 2016, Florida is likely to be a bit later in the calendar, and Nevada likely to be back in the first four, perhaps even #3, after Iowa and New Hampshire. That puts three potentially strong Paul states at the beginning of the calendar.
Other states where he might be able to do well include the AK, ID, ND, ME, MN, and ND caucuses, as well as the VT primary.