Their voice and commentary. As harsh as it may sound, if I can't stand the voice, then it's a no-go. Now, voices aren't something we can change, so it does indeed sound harsh, but that's reality. If you don't have a voice for commentary, then that's just the cold fact. So any LP's done by young kids or guys with high/irritating voices.. sorry, but no. Of course the commentary has to be good as well. It doesn't have to be funny of course, but it does have to be enjoyable to listen to. And genuine. Stuff like PewDiePie is just a pain on the ears.
They're the big things at any rate. Of course the game in question has to be interesting for me as well. If it isn't, then great commentary alone won't save it for me. An example is one of my favourite LPer's, veriax. He's doing a Dragon Age: Origins LP. The commentary is very great, but the game itself is just so boring, that I can't watch it.
So, those two combined is what makes an LPer for me. My favourites are the aforementioned veriax (mostly RPGs), Gopher (mostly Bethesda games) and ReformistTM (mostly Mount & Blade: Warband and its massive modding scene). I also watch a lot of PrinceofMacedon for Rome Total War stuff.
Edit: Also, I do enjoy a good blind LP. It's fun to see the genuine reaction from an LPer on something. I've watched several blind LPs just to see how people respond to certain events.