FanlessTech found quite a few models of small form factor PCs on Amazon and, while it's not listed in the specifications, some of the manufacturer Q&A responses state that they are fanless designs. Each of these devices are built around the Broadwell Core i7-5500U, but that might not even be the best part. Each PC has 4x USB 3.0, 2x HDMI, and 2x Gigabit Lan. Dual LAN and Dual HDMI opens up quite a few possibilities for a cheap, silent PC, especially since it has a relatively high-performance processor.
Prices range from $360 to $444 for 2GB of RAM and 8GB up to 256GB of SSD storage (with several models between). A single 8GB RAM model, with a 256 GB SSD, is also available for $483. They also found one with an i3 processor, but you need to bring your own RAM, SSD, and WiFi. It does have the same port layout, four USB 3.0, two HDMI, and two gigabit LAN, but might make more sense to grab the Core i7 versions unless you already have DDR3L RAM and an SSD hanging around (or 2GB is insufficient and the 8GB model is out of your price range). At $221 USD plus these components, you probably will not be saving much to compensate for the drop in performance. You can also find some Core i5 models, too.
Quite a bit to consider, but I think that many would benefit from the thought.