I've had a number of forays into the "fish world", with some success & some sad outcomes.
Betta fish are beautiful, fun to watch, & fairly easy to maintain. If you get one, do NOT get the tiny Betta bowl. They do breathe from the surface of the water, but they do much better in a larger container - 2+ gallons. It can be a large vase, a bowl, an angular-sided tank... They also appreciate having some gravel on the bottom, & a few large slate pieces or plants to hide in. Plastic plants are fine (real plants can be more difficult to maintain)
http://bettafishcaretaker.com
Goldfish are also fun, & you can get "feeder" fish for very cheap. That way, as you become more experienced at keeping fish, the out of pocket cost to your mistakes is lower :) They are another fish that can be kept in a bowl, but Goldfish tend to be messier than Bettas, & will need more water changes.
http://www.fishnet.org/goldfish.htm
Getting an aquarium can be fun also - it doesn't need to be big or complicated. A 20-gallon freshwater tank is fairly easy to maintain, and once you get it set up & established, you can create a community. Easy fish to maintain in these tanks include guppies, platys and tetras. A lot of people will get a pleco, there are a lot of varieties if you are willing to spend more $$.
Aquariums can be expensive if you get them new from a pet store, but you can often find a set-up for sale on Craigslist. They require some type of filter to clean the water (you will still need to do a weekly/monthly water change) and aeration (aquarium fish are not surface breathers).
Chances are, if you start with one fish & enjoy it, soon you will be looking to do an aquarium. Fish can be addictive, but that's a good thing!! And it can be a nice family activity, believe it or not. My husband & I had a lot of fun researching different fish to see which ones could co-habitate, and what type of environment they thrived in.
Enjoy your son's new pet! T. :)
ETA: Diane B - Bettas can live in a community fish tank, you just need to be careful to make sure they live with other fish that will not pick on them. Because they are a slow-swimming, ornamental fish, more aggressive fish are not compatible, & will attack their finnage. Also, they must be the ONLY male Betta in the tank, but you can also have multiple female Betta fish. :)