Besides the obvious problems with having the grip strength of a dead fish, a weak grip can cause elbow pain. If you can't carry a load equal to your bodyweight in each hand for 15 meters (a bit over 16 yards), your grip sucks.
A lack of functional grip strength not only limits your big-boy gym stats, but it can cause some horrific trauma to the deep finger and wrist flexors that adds up over time. You must improve your grip strength to prevent or eradicate elbow pain for the long run. If you think your grip is more than adequate, you're probably mistaken.
The Test: See if you can carry a load equal to your bodyweight in each hand for 15-20 meters. So if you weigh 200 pounds, you should be able to carry 400 total pounds for that distance.
Yes, that's pretty damn heavy! At the very least, a functional carry should consist of 75% of your bodyweight in each hand. If you aren't here, it's time to just get better at the basics.
To enhance elbow health while increasing maximal grip force output, increase the load on your carries in a linearly periodized progression. Keep the distance relatively short (10-15 meters) and treat your loads no differently than any other strength movement in your arsenal.
Program carries at least twice a week, making sure adequate rest periods are taken between sets. The goal is to increase your anaerobic capacity, not to walk around with 50 pounds in each hand for a half hour like you're a UPS deliveryman.