I think I liked the Eccleston Doctor the best, but I have a terrible memory when it comes to fiction and I don't remember why I like him.
I'll go with Pertwee. To me he brought more gravitas to the role. Tom Baker got too over the top campy in my opinion and with the first two, the show was too amateurish production wise.
Since you could only choose one, I went with the 10th Doctor. He seemed the most realistic to me, he showed genuine empathy and sympathy when he needed to, but also had a cold and dark side when he needed it. The second pick would be the 9th Doctor, he seemed to have fun all the time even when he was carrying all the guilt he had in regards to him thinking he caused the death of all the Time Lords, followed by the 4th Doctor so far. I haven't watched any of the episodes with the 11th or 12th Doctor.
Tom Baker because he was the first one I saw, plus he was so eclectic. Second would be Sylvester McCoy because he was Baker-like but with a more mysterious edge. Pertwee third because he was bad-ass.
Tom Baker was my first, and even having seen the rest of them, he's still my favorite. Sylvester McCoy comes in second - the perfect blend of impishness and deadly seriousness. Then Tennant, the best modern Doctor.
I'm a (Tom) Baker girl through and through, but i do admit that a lot of what was made back then was hopelessly silly - as far as the SHOW, the new stuff is way better. #2/3 is a toss up between Eccleston and Tennet. 4. Smith 5. Colin Baker 6. McGann (potentially, better than Baker and even Smith but it's hard to get a feel for him over all the "BOOM!" in that movie) 7. Davidson (underrated IMO) 8. Hartnell 9/10. Petwee/Troughton - just...meh. 11. McCoy - not his fault maybe, but in his first episode he confronted a supposedly deadly enemy with not one but TWO pratfalls. Sorry. i can't overlook that. But really, for me there's a huge gulf between the new show and the old one apart from Baker, in my heart Tom Baker was the only REAL old school doctor and none of the rest before McGann count at all.
Gotta go with #4. He was my first Doctor and his early episodes still hold up very well. Very well written, dark, and engaging. Probably the first actor to really make The Doctor feel like an alien and an outsider. Of course his tenure was during the '70s, so the later episodes did get a bit campy but still very good. Smith has done a very good job. Colin Baker got badly screwed. At the time I thought JNT was a very good producer, but in retrospect he was a much better art director or something than producer/show runner. As for Doctors that I relate to the most, gotta go with the First. Cranky, secretive, antisocial. If I were going to play The Doctor, he's the one I'd be. (I actually got him on a "which Doctor are you?" quiz).
Same result I got. To the point that a friend of mine recognized the description as me without knowing anything about Doctor Who.