Yes you need a filter. You need somewhere for the bacteria to grow, and the best place for that is on sponges or filter wool ect, inside a filter. The best thing to fishless cycle with is pure ammonia. And you really need a test kit, a liquid one for the best results.
But really you need a tank not a bowl, bowls need to be banned outright .
5 gallons minimum for the betta to have plenty of swimming room. That's not big, it would fit on a desk easily. Also the bigger the tank, the more water, the EASIER it is to look after and to keep the waste to a minimum. A filtered and heated 5 gallon will not only look better and be easier to take care of, but will be MUCH kinder to the fish and it will live longer . There have been reports of male bettas living up to 9 years, about 5-6 seems to be the average.
Facts to tell your mother. I mean actually show her this.
Although the bettas we buy in stores are bred to have long finnage, they still have all the normal instincts and bodily organs of a wild fish. With that in mind, I can tell you now that wild bettas do NOT live out their lives in tiny shallow puddles as many seem to believe . They die in them in the dry season. Some survive of course, due to the fact they are anabantoids and can take in atmospheric air. But to deliberately replicate ( in the form of a bowl ) the conditions in which the wild fish die in great numbers is pure lunacy and
immensely cruel.
The local people collect the males in the wet season, when the rice paddies and swamp streams are FULL. Yes they are shallow, but the environments as a whole contain millions of gallons of water. Compare that with a 1 gallon unheated, unfiltered bowl of death, and it's a damn poor lookout.
You need a filter. Why? So that you have a biological cycle. The bacteria in the filter eat your fishes waste products, and make them less harmful to the fish. After that you need to do regular water changes to remove the last of the waste and keep the levels low. When fish are left in their own waste, with irregular or no water changes, they will get diseased, they will die, they will basically suffer horribly. That is a fact.
You need a heater of some sort . Why? So that you can keep the water at a constant level temperature. The room temeprature WILL fluctuate. Doors and windows will be opened and closed and the temperature is bound to change at some point. A heater removes that problem. You set the temp on the dial and that's how it stays.
You need a test kit. Why? Because you need to test the water to see if the waste your fish is making, is getting too high. The bacteria in the filter can only eat so much. Not every last bit can be eaten, and so it will build up in the water and make your fish very sick if you do not do water changes. With a liquid test kit you can do various tests to see what is building up, and then you do water changes to remove the waste ( you should do that weekly without fail ) .
You also need a test kit in order to start off the biological cycle I mentioned before. You have to measure when the ammonia and nitrite get to 0 ( which takes about 1 month ) and then you do a big water change to get rid of the excess nitrates as well. Then you can add a fish safely and humanely without exposing it to the harmful substances, and the levels of bacteria in the filter will be enough to cope with the waste produced by the fish.
All of this will be easier to do, and safer and less stressful for your fish, in a tank of 5 gallons or larger. Many keep their bettas in 2-3 gallons and do very very frequent water changes. Not something I agree with in terms of space and ease of tank maintenence personally but if it works for them then fine.
But the basic rule is, the more water you have, the better you will be able to keep a stable tank environment and biological cycle for the fish because the waste is more diluted. Some say that bettas do not like so much space. That is easily solved with heavy planting ( real or soft silk false plants, real are nicer though, and use up waste too ) and plenty of hiding and resting places. Think about it. If you'd been kept in a tiny cup for ages in a store and then plopped into what seems like a vast space with strange objects and plants, you'd be freaked out too.
If your mother is still doubtful as to why bettas in more than 1 gallon do better, show her some of the ones in the Betta forum. There are so many happy bettas there, beautiful fish. Here is my mothers betta in his 5 gallon planted tank as a quick example. If space is an issue, I can tell you now that that tank is on a small table in the corner of the room. If we had a desk or dresser in there it would also fit on that.
I can get the kit, but she'd be extremely mad at me for getting it. (It's my money, go figure.) What can I do to make the fish happy? If I have to clean the bowl every day, I will. And if I have to make my mom mad I will, I'd just like to avoid a few fights. So, my main question is how I'm going to cycle a fish bowl. I'm pretty sure there was something on this forum about it, but I can't find it for the life of me. Geez, this would all be SO easy if my mother would just actually believe me when it comes to my pets!!! Just because my last betta fish lived for a long time doesn't mean it was happy or healthy! 
Hehehe.
