I think this is a good idea... as long as you don't make it intrusive.
This means that:
- it should not change regular use of the site
- it should not "stand out" in any way
- it should be aesthetically pleasing, even if not viewed as a puzzle
Unfortunately, mods don't have absolute control of the site design. We'd have to appeal to the SE overlords for every new page, feature, or design change we might add, and judging from our past track record with asking for exceptions, it doesn't seem too promising.
If we could pull it off, it would be nice, but I think there are too many restrictions for it to be workable.
(If you did find some places where information could be hidden naturally, I'd be happy to design a puzzle around them. I don't think we have enough leeway though.)
And remember, any changes we make would be effectively permanent. We'd probably have to have a whole suite of puzzles, since changing the site design for just one would not be worth it, either for developers or for us.