It's a pretty regular occurrence for me to have a friend, student or acquaintance come up to me and ask "I want to get in better shape, what should I do or work on." As most if not all of you know by now I prefer bodyweight and functional strength training above anything else. My most common recommendation is first joint mobility/flexibility followed closely behind by grip and core training. Now some of you may not think that those are particularly important, but take a look at your grandparents. As they age they start loosing mobility, balance and strength - most of which goes back to grip and core.
"But I'm a martial artist."
Great, so having a strong core and grip can help you on the mats as well as off. I have the same suggestions for new grapplers as I do with your everyday, out-of-shape guy. Ill say something that some will argue against, but oh well.
Grip and core strength will make you better at grappling.
This holds very true when newbies get on the mat, most if not all lack good body control along with the overuse of improper/over gripping. Will working these two attributes help you take someones back and choke them - No, of course not. What it will do is speed your progress in learning techniques because you will better be able to preform the movements properly and with less effort. This is especially true if you do sports specific grip and core training. Here is another point which every competitor will agree with me on - what is one of the first things to go in a gi competition besides your cardio? - Grips.
So hopefully I've provided you with enough information for you to believe me and if not, take it from someone else who will show you the in's and out's of grip and core work. Here is Lance from StrongFirst, showing us grip progression exercises for strength.