Just found this very helpful description of the boot process and what happens at each stage:
http://reviews.cnet.com/8301-13727_7-57345.../?tag=txt;title - I would agree that it's not the monitor or anything to do with the video system, since you can see the Apple logo.
This seems to be a distressingly common issue with the mini. There are quite a few potential solutions if you Google "Mac mini with white screen with apple logo" and variations of this.
Some more suggestions:
https://discussions.apple.com/thread/454134Replacing PRAM batteries is possible, but on the most recent models looks very difficult and requires special tools, according to ifixit.com.
http://www.ifixit.com/Guide/Installing+Mac...+Battery/3162/1