You can run it headless (no monitor) and connect as required from the iMac.
The obvious use would be to run a web service ... host a blog or website, or host an eCommerce site where something you decide to set up as a hobby can generate some income, etc.
Because the Mini runs cool and uses little power, relatively speaking, you can run it 24/7. It will happily serve as a security system (you can install it anywhere, including out of sight anywhere you can run an ethernet cable to) recording video from a number of cameras installed wherever you feel it would be helpful. The video can be accessed remotely with the right software, and notify your smartphone if you're away when an incident (you set the parameters) happens.
You can set up a Network Drive attached to the Mini and accessible from your iMac or other computers you may set up from time to time. Similarly, you can set it up to perform regular backups so that your data is relatively safe (to be truly safe, you need a second, removable backup drive that can be stored off-site, in case of fire or hurricane, etc).
Other World Computing have excellent external enclosures that have the same form factor as the Mini. You have a wide choice of configurations, from a hard drive alone to one incorporating a Blu-Ray recordable drive plus HDD. They look very nice and are an option if you will be setting the Mini in an area where it's visible rather than hidden away.
Having said that, the enclosures are not terribly expensive and offer many expansion options (Hub with extra firewire, USB, Lightning). You can explore the advantages of using Firewire to connect your music drive instead of USB. The short answer ... Firewire is self-managing, while USB requires CPU cycles. Firewire is much easier on the CPU which can then concentrate on serving your audio files and routing to a USB DAC.
You can set up a Gigabyte Ethernet network via the Mini, negating the need for a Gigabyte router. In the same fashion, the Mini can manage your wireless network, no need for a wireless router.
You can connect a printer to the Mini and print remotely to it (from your iMac, and from your smartphone) rather than having to find a place to put a printer in the living room. You can set up a station to scan photographs or slides if you have any from the pre-digital age. Any and all peripherals that you don't want to clutter your living room where the iMac is, can be set up somewhere else connected to the Mini.
You can use it instead of an Apple TV, sending video to your display. Your Mini can easily send HDTV signals to a Plasma or "LED" television. A "dumb" display works fine, no "Smart TV" required, as the Mini then does all the things a "Smart" device does.
We're only scratching the surface here. I guess it's a sign of the modern times, but it seems people have a hard time figuring out what a computer is good for beyond eMail, surfing, and storing music files. But the oldest computer is secure if you don't connect it to the internet, while a modern one like your Mini can safely expose itself to the 'net and still function in a number of useful ways.
I understand how some suggest selling it; it will never be worth more tomorrow than it is today. But a few hundred dollars is the most you will get and you can't replace it with a new machine with similar power for anything near what you would recoup if you did sell it. Personally I would try to find a use for it and give it a job.
