The Doctor shouldn't be anyone famous. It definitely won't be Benedict Cumberbatch (who has said that he wouldn't play the Doctor, although it sounds as though he'd be open to the Master). It won't be someone high profile like Helen Mirran. And while I do think Ben Whishaw would be fantastic, I wouldn't necessarily cast him either.
I do think the Doctor should be an unknown. Like Eccleston, Tennant and Smith, he should win us over as the Time Traveler from Gallifrey, not as a famous personality.
(okay, side note, I started watching Doctor Who partially because I'd thought Tennant was cute in Harry Potter and wanted to see more of him.)
Except... there is someone I really, really, really want to play the Doctor, and that is Bill Nighy. He was actually in consideration, although I don't know the specifics of how serious that was. He did get an uncredited cameo in "Vincent and the Doctor" which shows he is supportive of the show. And he likes doing smaller, funky projects. And it's high time we had an older Doctor again.
Seriously. This man was born to play the Doctor. Forget the CGI performance of "Pirates of the Caribbean" and go watch him in "Wild Target" (Netflix) and "The Best Exotic Marigold Hotel." He is brilliant. Quirky, full of hidden depths of emotion, serious yet able to show the joy of life... I might seriously cry if I never get to see him play the Doctor during my lifetime.