I don't know. I usually substitute puns for wit.

The brain is an "associative machine" - it does pattern matching, imprecisely. In my experience, if I listen closely to the babble of my brain, it's trying to understand by seeing if stuff fits this meaning or that, using all sorts of information, like the way the words sound and rhyme, the cadence, as well as ideas associated with the words. Sometimes when I hear these sub-thoughts, I laugh.

I'm no expert on this, but I think wit is attending to these imperfect matches and sharing the ones that are funny, that make you laugh. I don't know if it can be developed. I'd try meditating and smoking dope (only in a state where it's legal of course, or in a state of complete disregard for the law.)
And, it may be partly a genetic difference. What's valuable about people is they can be "present", they can be aware of surroundings and what's happening, including listening. As someone who enjoys other people's wit, you are as valuable as they are.

Most people downplay their own value and think others are more value. The exceptions are assholes. But yes, it's nice to say something funny now and then (even better are those rare occasions where it's on purpose!)

And, try watching and reading comedy. Maybe some of it will rub off. You can also read jokes and practice saying them when you're by yourself (between the sheets). And tell your friends that you'll be trying new ways to be witty. Tell them it'll sound stupid for the first 100 years, but after that you promise there'll be improvement.

At times I've been witty. But often my wit is just annoying. So you have to be brave, and not take yourself seriously. If you're serious about being witty, you've missed the point (of the joke). Be like a kid, have fun with it, play. But do it with friends at first.