It's been my experience with other forums that a minimum search length of 3 characters is common, and is the default with phpBB. This forum software, XenForo, has a default of 4. Meaning anything 3 characters or less won't return results like you're seeing. For words that are too common, that has to do with a certain threshold that once crossed flags that word so it won't return results. In both of these cases, changing these limits will increase the size of the search index files, and can potentially make searches take longer. Going from 4 to 3 for minimum character length probably won't cause any noticeable difference, but going from 4 to 1 almost certainly would. I think that the percentage for the common words threshold has a pretty negligible effect, going from something like 5% to 25% for example.
I'm not bothered by the limits personally, and just think you might have to get a little more creative with what you search for, and maybe expect to spend a little time digging. Just posted some tech info because it could be done and shouldn't cause any issue, but probably doesn't need to be done.