Re-encoding a video using the same settings would certainly degrade video quality further; audio can usually simply be copied over.
According to this Handbrake page, that option adds "an atom" to the file. More detailed information seems to be hard to come by, so I believe your best option is to re-rip from the original source.
Alternatively you could try these methods:
Re-encode the file with Handbrake
Make sure to copy over the audio tracks so they don't degrade as well. This is accomplished by selecting the appropriate (or auto) passthru audio encoder on the "Audio" tab.
Try importing and converting the file in iTunes as per the post on this forum thread:
I assume your movie(s) are already in iTunes; if not then import them.
Select a movie, then in the menu bar select Advanced / Convert
selection for iPod/iPhone. This leaves the movie unchanged, but
creates a new version compatible with iPod.
I can't say for sure whether the video quality would degrade using this method, but my guess is yes.