First, see if you AVI can be played using QuickTime Player. If it will, press the letter i while holding down the Command key. This will open the Inspector. Tell us the dimensions of the clip and everything you see next to the word FORMAT. We can give you a better answer with this information.
Having said that, a very good converter, that also happens to be free, is called MPEG Streamclip. You can download it here. Get the Beta version.
Drag your clip into MPEG Streamclip. Then make sure it plays OK by clicking on the play button.
Then click FILE/EXPORT TO QUICKTIME. Choose Apple Intermediate Codec for the compression, and save it where you can find it. If this works, you can import this file into iMovie using FILE/IMPORT MOVIE in iMovie.
If it doesn't work, give us the information requested above.
Download the free application MPEG Streamclip from the developer's site here:
Drag the AVI movie onto MPEG Streamclip then click the Play button to ensure that it plays OK. From the menu select File > Export to QuickTime. For Compression select Apple Intermediate Codec. Leave other settings as they are. Now click Make Movie (bottom right of screen). If desired, rename the movie (or leave it as the same name as your project), then Save to Desktop or your preferred location.
Apple Intermediate Codec is iMovie's preferred editing format (also referred to as AIC). The exported movie should work perfectly in iMovie.
EDIT: Drat, beaten by AppleMan
Unfortunately i am still stuck. I have lots of AVI clips taken on an older Flip. When i drag the clips onto MPEG Streamclip, nothing is there, all white screen. My mac is brand new, so not sure what the glitch is. They won't play in Quicktime either, but they used to play when i was running Snow leopard on my old mac.