MobileOrg[1] was good for a while—it had Dropbox and WebDAV support—but it hasn’t had an update in a couple of years and is badly in need of a new maintainer.
Recently someone has made a new client called Orgzly[1] which is actually really well designed. Sync works well enough for me over Dropbox, though I'd really rather use ssh. I guess overall there's a little impedance mismatch between Org-mode plaintext and mobile interfaces but overall I think it works well.
Dropbox support was cool, but I recall MobileOrg being buggy to the point of making it completely useless, not to mention pretty bad UI. This project really needs some love.
[1] https://play.google.com/store/apps/details?id=com.orgzly