The idea is to take the heat out of your skin so holding a burn under cold running water will cool it down quickly. Your mother's suggestion will certainly melt the butter but it won't do your burn much good.
I have always had really good luck using mustard on minor burns such as from grease pops or even cigarettes - sounds weird i know - but the vinegar works really well to not only relieve the pain but remove the heat from it as well. Just dab it on the burn and let it sit for a while - but just so you know - cold water works too!
A tip I got from a nurse: immediately after you burn yourself, gently rub the burn with dry (preferably unperfumed) soap. I have no idea why this works but I've tried it several times and swear by it.
My daughter says to wrap a burn in cling-film if it's handy. Doesn't stop any pain but keeps it sterile till it can be treated properly. I'm sure you could run cold water over it with the cling-film on or wrap a wet towel round it too.
Be careful. Burn heat starts to cook the area burned, Butter helps it COOK. Getting the temperature down is important. Run the burned area under cold water for about 15 minutes. Any less leads to blistering and marking. For serious burms get help immediately.