My family needs one solely for watching video streams. It will be connected to the main TV in the living room. No need gpu or os. Chassis wise, smaller the better. What's the cheapest this can go?
If you don't mind a less pretty and bulkier case (with a noisy PSU).
This is probably the cheapest HTPC you can build:
Asrock H71M DGS - $69
Celeron G1620 - $55 (get it from cashtronic)
G Skill 1333 4 GB - $48
Vios mATX case + noisy psu - $55 (Laser Distributor)
8 GB Sandisk Cruzer - $8
Total: $235 (prices from PC Themes)
Optional: Prolink Pan 2001 USB wireless adapter - $19.
Look for ralink or atheros chipsets as they will have a higher chance of being supported.
Install openelec XBMC. Can spend the remaining amount on a remote
Alternatively you can take others up on their suggestions, but you can always replace the HDD with a small capacity thumbdrive with openelec installed.