Half-height cards are cards that are roughly half the height of normal PCIe cards. They're for use in smaller computers that for whatever reason are built to use half-height cards to save space.
As for what card to get, it's entirely dependent on what your coworker wants to do. Are they wanting to game? Or just watch movies? Etc. And how much are they willing to spend?
You don't really have a lot of choices since the half-height form factor limits the cooling options (and hence overall performance) of the cards, but there is some difference.
As far as your most modern options are concerned, AMD's bottom-two cards in the Radeon 5000 series - the 5450 and 5570 - are both half-height. There are also older low-end cards from the last generation that tend to be cheaper, and there are a few one-off custom designs such as Sparkle's half-height 9800GT which attempt to squeeze a more powerful card in to the form factor.
I'd suggest starting your search by looking at what
Newegg offers. They have their half-height cards organized as being "low profile" and "low profile ready", so it's easy search by just half-height cards.