There is a very generic rule about fish and tanks (and I do mean generic, because obviously, it does not apply to all fish) that indicates that you should have an ABSOLUTE MINIMUM of 1 gallon of water per inch of fish (again - this is very general, obviously a 10-inch fish isn't going to fit well in a 10 gallon tank). That being said, most Betta fish get to be 2.5 inches, so if you follow this rule, you would need a minimum of a 2.5 gallon tank when he gets bigger.
However, you can make a 1 gallon work if you clean it often enough. A filter in that size tank can be a bit overwhelming for a Betta, so, you would probably be better off without it. You should clean a 1 gallon tank a minimum of 3 times a week with one of those being a 100% water change if you don't use the filter or have live plants (if planted heavily enough, live plants can replace the need for a filter).
Here is a helpful link from this forum that talks about how many water changes you should do based on your tank size and whether or not it has a filter or not. As you will see, the bigger the tank, the less water changes you will need to do. http://www.bettafish.com/showthread.php?t=115758