I am assuming you are talking about in a public situation or a party. Either way, begin by diffusing the tension. Smile and make a joke that is tasteful and will seem funny. Even if it's not funny, most people will laugh and appreciate that you are trying.
If you're talking about introducing yourself to a group in a public speaking venue, it would depend upon the purpose of your speech. If it's casual, you might start with a humorous story about yourself. The story should be relatable to your topic and your audience.
If you are talking about introducing yourself in front of a group, maybe you could start by talking about something that sort of defines who you are. You might talk about your favorite song or your favorite hobby. So, instead of saying "Hi, my name is..." you could start out by talking about what this song or hobby means to you. That will tell your audience something important about you. After that, you can go into things like your name.