Mostly use as a hobby,interested in what I can do and would like to have family members reach my server. post personal photo site for family and for family reunion.
Making it a remotely-accessible web server, available for members of your extended family, is not going to be very realistic. It is technically possible, but requires some customization of your network hardware. (Not all cable or DSL modems may be compatible with this.) It will require that your internet service provider (ISP) assigns you a static IP address, which most ISPs will charge extra for. (Without a static IP, your address may change, requiring you to tell everyone what the new address is every time.) Alternately, you could pay for DynDNS to give you a domain name for a dynamic (ie, changing) IP address. And once you have all this in place, if your server starts to get too much traffic, your ISP may notice and cut you off. Running a public web server is not something that the average ISP will allow without significant additional payment.
That said, you can do a lot on your local network with the Mini. Hook up a big external drive and use it for Time Machine backups for all your other Macs. Put a music library on it in iTunes and share it with all your other computers via Home Sharing. Get an Apple TV to stream movies and music from the iTunes library on the Mini. (Check the requirements on that, you need a certain minimum version of iTunes for streaming to an Apple TV. Also, that becomes much more enticing if you rip your DVD collection and put it in iTunes on the Mini, which is the only way to wirelessly stream such video to an Apple TV.) Put important files that your whole family needs to share on it and allow the other machines to access it through file sharing. There are many possibilities.
