Probably what your friend is experiencing is a lethargic betta due to cold temperatures. Bettas are tropical fish that require a steady temp of 78-82 degrees F to be happy and healthy, and therefore your friend will need to purchase a heater unless she has a room in her house that stays a steady 78-82 degrees.
With that said, there really is no way to heat a 1 gallon without cooking the poor guy, so your friend would need to upgrade to a 2.5 gallon or larger....5 gallons+ is recommended for beginners. The larger the tank=less maintenance because ammonia(which is harmful and even deadly product of fish waste and uneaten food)will not build up as quick in a larger volume of water.
If she really wants her fish to be more active and start showing more color, she will need to upgrade to at least a 2.5 gallon(again, 5 gallon+ is better)and at least get him a heater. A filter is not necessary, though without one that means more water changes. In a 1 gallon, she should be changing all of the water every other day at least, like suggested, to keep the ammonia in check.
To add my view on the subject: Many will say that a 1 gallon is fine for a betta, but I honestly believe that using something so small for anything more then temp housing or a hospital tank just isn't right....They are very active and intelligent creatures, and need more room then that to really be able to stretch their beautiful finage and be active to their full potential.
Hope this helps some ;)