Hello: I'm making a trip to London in January, and I plan on seeing:
Antony and Cleopatra
Pinter at the Pinter
The Mousetrap (a tradition for me)
True West
I have spots for two more shows. I have some candidates: The Inheritance, A Very Very Very Dark Matter, Macbeth (Globe), Summer and Smoke (Almeida), 42nd Street, The Comedy About a Bank Robbery, Foxfinder.
I'd avoid the globe as January will be cold so open air theatre doesn't sound like too much fun. Foxfinder isn't getting too favourable buzz so I'd avoid that too. The Inheritance is an incredible piece of theatre, but surely a New York production is likely so you might catch it there (not sure where you are coming from). I'd probably pic a Very Very Very Dark Matter (if you are a fan of Martin Mcdonagh and his dark comic ways) and then 42nd Street if you want something glitzy and cheery or Summer and Smoke if you want something a little less jazz hands-y.
Obviously Macbeth will be staged at the Sam Wanamaker Playhouse of the Globe and not the open air theatre, since it's closed for the winter. In case you haven't been there before, it's a lovely reconstruction of a Jacobean playhouse. Not very comfortable, but really suggestive.