f you want to build a community tank then by all means buy a 10 gall and cycle the filter. however...if all you want is a betta, the idea of you being influenced by existing members to buy a 10 gall and cycle the filter is laughable. yes, it is better to cycle a filter becuase this helps with the nitrogen cycle, however, this is not needed with one tiny little betta. Even if you follow the '1 inch per gallon rule' which is not particulaly accurate, then you will still have say 8 gallons spare....and considering your mom is concious of cost etc is a silly idea.
I used to keep my betta in a 6 gall cycled tank with filter/heater etc. it lives in a 3 gall still tank now and it is much happier, builds bubble nests daily, colours are more vivid and the fish is friendlier
you don't need to go out and fork out a shead load of cash so members will not percieve you as 'cruel' the concept is ridiculous