LyX participates in the Google Summer of Code 2013
latex-community.org
latex-community.org
This means having an API and usable connections to existing online services. I'm sure the vendors of such services would support it. So a user could simply install LyX, and benefit from the newest TeX version without installing it - slim but capable and up-to-date installations would be possible.
File storage services such as dropbox could be connected, the storage of an online compiling service used, or files could simply be uploaded each time when being compiled - LaTeX files are small, and compressed uploads or syncs would be quick.
Edit: I should also point out that there is a fairly well-established protocol for online latex compilers, namely that of the CLSI (https://github.com/scribtex/clsi), which is maintained by jpallen at ShareLaTeX. So, it should be straightforward to do this in a vendor-neutral way.
Oh, c'mon. Link it at the end, or even at the top, but with a "this post is also being discussed on Hacker News" or similar anchor. Don't blatantly pander for votes.