Short version: I ditched my network security.
I'm one of those iPad customers experiencing wi-fi issues. Apple has acknowledged the issues and offered this solution and other solutions have been compiled here. Unfortunately none of these worked for me.
Moving my router to the middle of my house did help signal strength, but it still kept asking for a network password intermittently and especially upon wake up. Often when you then typed in a network password, it would then reject it as incorrect, which was just plain weird.
After iterating a lot with my network security and other settings, I finally decided to try no network security, which pretty much immediately fixed everything. Now maybe it is still cutting in and out and just not asking me for the password, but if so, it's doing it in a way I'm not noticing.
Of course having no network security is not a good thing. Thankfully I live in an area where it's really a non-issue. Nevertheless, as it seems this may be a software issue I'm hopeful it will correct itself upon some future Apple update.