I used to install CoLo boxen for clients, sometimes dozens in any given data center. Unless you have network KVM or its equivalent, you'll end up hating the idea. And you need to plan for backups too.
With a mini tower, you'll most probably have to get a shelf and pay for at least 3-4U if you lay it on its side, more if standing upright. Since stuff is always rack mounted, cooling can be an issue as well. Rack mount servers are designed for that environment.
Since you are already using EC2, why not dial up a couple of micro instances. If you are running Linux or BSD, then do your numbers. Sometimes it works out cheaper to host multiple processes on a single larger instance.
