I have a seasonal cottage with 30 year old 1/2' pipe gravity feed water system. I now have a leak under the kitchen sink. I am wondering if I should use the same black abs (I presume, don't remember from back then) that was previously used.

ABS is not permitted in many States. Use CPVC. Inexpensive, User friendly, a DIY dream.

I agree with Mao64, I would replace with PVC pipe. CPVC isn't necessary for cold water. Good luck. Tom

PVC is a good choice for cold water. CPVC if you have hot water.