I play a sport where there is a huge advantage to players who can play equally well with both hands. Is there anything I can do inside or outside of the gym to improve the coordination of my weak (left) hand?
Going both ways doubles your chances of scoring in most activities 
Seriously though, spending a higher percentage of your time working on your weak side will help the most.
If you need to improve hand eye co-ordination juggling, and playing two handed catch (you can throw a ball against a wall as another is coming back to you, or throw two ball back and forth with a partner) can be helpful helpful.
Scratch DJs swear by those hand exercisers (Gripmasters) to develop their weak hand. Its easy to just walk around with one. It will build finger dexterity and forearm strength. I know it sounds weird to take training cues from DJs but they need to have ambidextorous skills.
dude are you kidding? isnt it obvious? spank with your weak hand, duh.
Right on P-DOG! I knew it would come from you.
Billy
There are many ways to accomplish this. Some are a lot more complicated that the others but certainly med ball work, throwing and catching a ball one handed will help tremendously.
As I said - there are many different ways but I hope that helps.
In faith,
Coach Davies