Saving disk space, but not quality
There isn't a perfect answer to this problem.  AVI and MP4 are just container's for video streams, so without knowing more about the actual streams in the containers, it is impossible to tell how much quality loss there would be, but as a general rule of thumb, it isn't all that atypical for AVI to use far less efficient video compression algorithms than MP4 on average.  That doesn't always hold true, but it often is.

That said, what you plan to do with the files also matters.  If these are a master intended for being a high quality copy you can come back and WATCH, then running a high quality h.264 encoding isn't such a bad thing.  You should likely be able to find a setting that will make the files quite a bit smaller and still be pretty high quality to watch.

The problem with this, however, is that while the compression may work well for watching the video, it doesn't work so well for editing and re-encoding the video later.  If you compress too much, the video will look great to a viewer now, but may look like crap after editing and re-encoding a final video later.  If you plan to edit the videos later, you will need to stick to a much higher file size and higher quality encoding.  This could still be h.264, or any of a number of other formats, but it would need to be a much larger file (such as all I frame h.264).

Personally, I've always elected to keep my masters at their highest available capture resolution and fairly minimally compressed, but I also am in a financial situation where buying large amounts of external storage is not a problem.  (I personally have around 20 TB worth of storage, of which about 14TB is full, though some of that is providing redundancy.  That's primarily for video projects, photo projects and Steam downloads.)

Ultimately, however, it is a personal decision about the trade offs between cost and quality.  Every option has it's costs and only you can decide which costs are best for you.  I'd encourage you to buy more storage if you want to do editing with the videos later, but if you are only worried about playback, I'd recommend encoding it down to a quality that you find high enough by trial and error.  (Also, be sure to check the videos as the level of activity in a video will greatly impact the video quality unless you are using a quality based encoder.)