I have no experience with instant cameras but am trying to choose one as a gift. What should I consider? I want it to print photos that are at least larger than a business card and of good quality.
Fuji makes and sells the Instax line which as you know is commercially available.
There is also a group called the Impossible Project which makes film for vintage Polaroid cameras.
Polaroid as a camera and film manufacturer is around in a different form and still making cameras as well.
There are a few specialty products floating around as well.
If you want something where film can be bought in a store or online, and the camera is new in a box, as a gift for a casual user, Fuji is probably the way to go.
I'd look at the Impossible Project just to see if anything there looks interesting (maybe the Spectra film.)
Updated: I totally misread your question at first and wrote an answer about compact cameras and sensor sizes. Look back at the edit history of this answer if you're interested.
What are some options if I want it to print photos that are at least larger than a business card and of good quality?
The largest instant media I've seen for digital cameras that you'd want to carry around is the Fujifilm Instax Wide 300, which prints photos that measure 86x108mm, or about 3.3x4.3in. That's roughly twice the size of a business card. I haven't used it myself, so can't speak to the quality.
There's an obvious relationship between the size of an instant camera and the size of the photos it can produce. People want small cameras but large prints, and that's a difficult thing to pull off in a single unit. I think you might be better off if you separate the imaging and printing functions: consider buying a compact digital camera (or just a phone with a good camera) and a portable photo printer. For example, the Canon SELPHY CP1200 is easily small enough to carry around in a backpack or camera bag, prints decent borderless 4x6 photos, can print wirelessly, and can run on a battery pack (not included, though). That seems like a good combination of print size, quality, and portability.