the lady at the Wal-mart said not to change the betta's water every-day, only when it gets murky.
The lady at the Walmart is a major idiot. No, really. "Murky" water = sick fish. Or dead fish. The end.
Now, I keep one of my fish in a 1.5 gallon tank. he's a major fin biter and a weedy, sickly little runt and yet he's going strong in the small tank. If 1.5 gallons is what you have -- it's what you have. Just make the best of it with adequate water changes and maybe add some java moss to help keep the water nice. And in a small tank, you'll need to do at least 3 changes per week.
My schedule for the 1.5 gallon is: 1 x 50% change, wait 2 days, 1 x 50% change, wait two days, then do 1 x 100%. Rinse, repeat.
It's not a lot of water, less than a minute in the shower's worth per week -- but if the changes bother your Mum, you could suggest that you can use the old water to water the house plants or garden -- the plants will love the extra nutrients!
Often, mild cases of fin rot (which he got from being stuck in the store with Murkwater the Idiot looking after him) will clear up with just really clean water and a good water conditioner like Prime. Prime is REALLY economical (I am on a fixed income, so price and economy is important) at 2 drops per gallon, and goes a long way compared to the really cheap, weaker brands.
You WILL need water conditioner. You WILL need a heater. Total price? Um.. about .. $25 bucks. That's what you need to keep your pet healthy and happy. Please let your Mum know that she is wasting her money and setting you up for a sad fishkeeping experience by not providing you with the basics for a tropical fish species...
I wish you luck, and hope things work out for you.
ETA: sorry, can't recommend a brand, I'm in a different country and not all your products are available here.. I'm sure someone will suggest a good, inexpensive heater.