I'd go with a mini ITX build with an i3/i5 to be honest. Running on Ivy bridge. With networked projectors, fairly sure a cheapo board should run 2x projectors and a screen. If not get a cheapo graphics card.
4GB of ram, buy a 128GB M4 and a 450W corsair PSU. Along with a singular nice silent fan.
Maybe a non-stock cooler to keep noises low when the machine gets filled with dust? Quite obviously no LED's
External HDD? You don't want to get a virus or anything on the machine and lose everything! Maybe get an old laptop drive and a USB enclosure for backups and a 160GB HDD for Internal? Depending on your needs you may want a soundcard too.
To keep the cost low i'd stick with a base i3 model and maybe even 2GB of ram. Get the samsung 30nm stuff and clock the crap out of it!
I know my ex's church had a build from a local PC shop, it's just an i5 and 4GB ram.Edited by Matt-Matt - 5/8/12 at 11:44pm