I've seen various posts with people needing to heat small tanks and other people saying it can't be done safely and to get a bigger tank.
Well I've got both my bettas in one gallon tanks and before I got my heaters when the weather started getting chilly the water was getting to a dangerous 55 degrees. Well I got a 25 watt heater for each tank and ever since the water has stayed exactly at 78...I haven't had a single problem. The heater turns on and off by itself and I love it.
Because in smaller containers the heat can build up quickly and toast your fish, so most people never keep heaters in anything smaller than 2.5g. It should be fine with you as long as you moniter the temp, but getting him a bigger tank would be best, for his own happiness and the ease of less water changes. The heater takes up lots of room in a 1g, so bigger is betta! (LOL, see what I did there?)
Yeah, I know it's weird but I also put my 1 gallon tanks on a seedling heating mat attached to a temperature controller. I was really into starting seeds last year and I bought those seeding mat with a temperature controller. So I set the temperature to 81F on the controller and put the probe into the water. The probe is water proof since that part is usually used to put into soil. And every time the temperature goes below 2-3 degrees of the set temp, then the controller will turn on the heat pad and turn off the heat pad when temp is reached. 81F may seem high but I also put one of those standing aquarium glass thermometer into the tank and that usually reads around 78F when the digital controller reads 81F. I'm not sure which one is more accurate because they're not calibrated or anything.
At night time it gets really cold in my room like 60sF (yes, I know temperature obsessed. I was really into orchids and so I had to monitor the temperature and humidity in my bedroom. I got one of weather temperature reader.) I also put a towel on the tanks at night so heat won't get lost so much. By the time I check the tanks in the morning it's around 77F. While the room temperature hits around the low 60s. For me that's really cold, since I live in Southern CA.
Well, I hope it gives you some ideas. I wish I could get a bigger tank also but fiance says no. And I already have a 10 gallon and 20 gallon tank. Yeah, I guess my new obsession now are bettas.
Being that your your house is cold enough that the fish tank got into the 50s it's no surprise that a 25 watt heater would work nicely in a 1 gallon.
However, generally speaking most people keep their homes somewhere near the low 70s so with that temperature difference you might find that a 25w heater would be too powerful for a 1 gallon tank.
Like someone else mentioned the problem with heating smaller tanks is not that there aren't ways to heat it, its that that small body of water will change temperature much more quickly than a larger tank.
For instance, say you have a 1 gallon and a 5 gallon tank filled up with no heaters. Lets say the tap water runs 80*F so starting out both tanks are at 80*F, you let them sit, unheated in your house which is set at 73*F.. the 1 gallon will lose heat WAY more quickly than the 5 gallon will.
So the problem becomes having a heater that can keep up with the constant changing temps of a 1 gallon (because as soon as the water is heated up it begins loosing heat the second the heater turns off), but that won't over heat the tank.
In my 1 and 2 gallon tanks I keep them warm with hydor mini-heaters. My apartment is kept cold (65*F range) and these heaters keep my 1 gallon tanks at a toasty 82*F, and my 2 gallon tanks at an acceptable 77*F. But if the heat comes on in my apartment (which I don't allow LOL) those 1 gallon tanks quickly turn into boilers if I don't watch out.
One option which I've never tried is the Marineland Stealth 10 watt heater. Its adjustable so it might be a good option for a 1 gallon tank. I've never read about anyone using one in a one gallon tank though.