I'm usually going for the 2nd option, using Rails to build the API, and backbone for the JS stuff. You can even get an admin panel for free using ActiveAdmin.
I've shipped tens of mobile apps with this kind of backend.
However it heavily depends if your app is interactive or not.
I did a presentation on this approach at the last RubyDay.it: http://www.slideshare.net/matteocollina/enter-the-app-era-with-ruby-on-rails-rubyday
For the third option, in order to get responsiveness of the 2nd one, you might want to try pajax as Github does.
