How often should I be doing water change for my betta? My wife seems to think it should be done pretty often.
The water change should help with some problems:
First, the waste of the fish itself ("poo and pee"), which will accumulate in the water.
Second, other waste, like uneaten food, dead plant parts and so on.
Third, the new water brings new oxygen, important for the fish to breathe easily.
The impact of these three points is different with every setting of your tank. In a big tank, it takes more time to reach the toxic concentration of waste or to breathe all oxygen away than in a smaller. If you have some plants in the tank they would influence it positively, because with enough light they will use waste as nutrition to grow and produce oxygen. If you use additional equipment like a pump with a filtration system, then the waste will be taken out, and if you let the water from the pump drop into your tank this will bring new oxygen into it.
In any case, you should regularly do water tests, because it could happen at any time that there are high levels of waste that the plants or the filtration pump could not manage. (For example, if the filtration material in the pump is too dirty and needs change/cleaning.)
Long to short: If your betta lives in a glass of water, yes! You need to change the water very often.
(I have to assume the worst case, until you give more detail.)