Handbrake doesn't really edit, but it does a great job of converting video files. You might be able to go into AVI preserving more quality using Handbrake's options.
Of course, you're talking about phone video, maybe the quality just sucks and you're noticing that when you view it on a bigger, higher resolution screen.
It's actually 1080p, and even when viewed on a bigger screen it looks very nice.
Are you sure there isn't another way other than converting? If there is I'd rather keep that quality.