Before I say anything about Mac Mini's, I want to mention that you can rent a dedicated, fully managed server for around $200 a month (fully managed doesn't mean they touch the software or O/S; it just means if something goes wrong with the hardware or the network, on-site engineers will fix it immediately; an invaluable service!). I've been using LiquidWeb (Google it) for my dedicated servers for years now. They fully own their entire data center and their support is superb.
If you're intent on using Mac Mini's, they will work just fine as little servers. In fact, there's a company who exclusively hosts Mac Mini's in a data center in Nevada (Google 'Mac Mini Colocation'). You configure the Mac Mini, ship it to them, and they hook it up. You get to own your hardware and have it hosted in a secure data center environment with lots of bandwidth. Of course doing this would also mean you no longer have physical access to the hardware.
Con's of using a Mac Mini: From what it sounds like you're trying to do, the con's of using 'non-server-grade' machines from a hardware perspective are not many. More importantly, however, will be the reliability and bandwidth of your Internet connection and future migration of your application to a data center and/or better hardware when your application begins to demand it. While building your own server and hosting it yourself will save you money, it won't save you time or headaches down the road (and hosting is really dirt cheap, especially VPSes; see below)
Other options you should consider with regards to cost (if you haven't already), are Virtual Private Servers (two companies I've heard great stuff about are SliceHost and Linode). You get a slice of a server and only pay for the memory, disk space, and bandwidth that you actually need (and VPS 'hardware' can be upgraded with a single click). You get root access and can choose what O/S you want installed.