Talk to your artist. Ask his ideas. Start with a couple words. A phrase. A situation. Use word associations. Use a creative process called mapping. Write down words and connect them when you see relationships. Look for universal truths and give them a funny twist. If you are creating the art yourself, consider your strengths and how you might best pair them with language. The gag-a-day comic stands on its own and changes every time. The story arc comic has a funny story line that may extend from a week to as long as 13 weeks with some serials.
Consider your audience. You'll want to gear your humor toward their interests. An office joke isn't for children. Some cartoonists rely on sight gags that are funny because of the situation they portray. This is common in single-panel comics. Multi-panel comics have two to four panels to set up a "straw man" and deliver a punch line in the final panel. Try introduction, build-up then punchline!
Review time-tested, popular cartoons, such as Mother Goose and Grimm, Peanuts, Garfield and Dilbert. Look for common themes within their humor, and consider how you might adapt these ideas to your own work. Keep a notebook just for gags and funny topics. Add a sketch to your joke if it comes to mind. Review your joke book often. Look for news events and situations in the headlines that you can give a humorous twist.
Look for inspiration in your own life. Consider events or people that make you laugh, and try to adapt them into a comic form.
Ask friends and family for ideas. Try out jokes on them.
After you've written a few gags, take some time away from the project. Come back with a fresh set of eyes and review your work. Make tweaks as you'd like.
Retry the process if you're unhappy with your work.