If you go the weight route, make sure you don't just train grip strength (wrist flexors) but also balance a bit with wrist extensor exercises or you'll just move the pain around. It can be as simple as wrapping a rubberband around your fingertips and expanding your hand a few times.

I found this balance issue important when I started working on my deadlift and my grip got a bit out of control. Also, if you have wrist problems, you might consider avoiding alternated grip deadlifts and just pulling hook grip.