Hello there,
The Gemfile helps to understand your current situation and of course the site link. As you know, your running Rails 3 with Mongoid 2. In terms of upgrading, would you like to upgrade to Rails 4 and Mongoid 3?
I have actually been doing a similar piece of work on 3 web apps I built, all were on Heroku's Bamboo stack running rails 3.1 or 3.2. I upgraded them to Rails 4 and since they were all running Mongoid, I also upgraded that too. So I literally have experienced this process during the last couple of weeks.
Since I already have a Heroku account, you can easily grant me access to your Heroku repository and then I don't believe it would take more than 3 days to complete.
I look forward to hearing from you.
Many thanks,
Daren