Everyone will probably say the sonic screwdriver, although I’m not sure it’s ever actually been used as a screwdriver. Yes, it’s pretty versatile and useful, but it doesn’t make me want one.
Some people might want to say the TARDIS, but, frankly, to do so after The Doctor’s Wife would be so wrong.
I’m not sure that Bessie counts as a gadget, but it’s too long since we’ve seen her and I’d like to see her brought back and given a bit of a 21st century update. That should be Moffatt’s Mission for this anniversary year.
The most useful thing, of course, would be psychic paper, accompanied with the Doctor’s seemingly endless capacity for bull***t.
It’s a pity this is just Doctor’s gadgets – and I’m not saying he couldn’t use hallucinogenic lipstick, but it does belong to River. That would be pretty good to have in your back pocket, though. Not much use against the daleks, but if you can get an entire legion of Romans thinking you’re Cleopatra, it must be worth having. And it even has its own thread on Tumblr.