Try asking the 14 year old birthday boy/girl what they would like to do. They may surprise you and come up with a cheap idea. Parents today are far too busy today tryng to outdo each other. Make sure the teenager has fun and your time because that is what they really want.
A good thing for teenagers is ten pin bowling,you can get rates for birthday parties, and boys and girls are equal,my son loves it,where do you live, and how much you want to spend comes into it,longleat,lego land, picnic on the beach,loadfs to do cheap and lots to do of you want to spend the money
have you asked her if there is anything she would like to do? Does she want a party or just an outing with her friends?also what is the budget? Maybe as a 14 year old she would appreciate a little trip to the beauty salon to have her nails done or a makeup lesson, perhaps with one or two friends, and then a meal out afterwards? maybe she would just like a shopping trip to buy a new outfit and again, a bite to eat afterwards. Is she sporty? would she enjoy going skating with friends or a party at the local swimming pool? tickets for a theme park, Alton Towers or similar? Sorry am not very helpful, my daughter usually just had a party or as you mentioned , cinema or bowling. Yr daughter is a lucky girl to have such a thoughtful Mum. Le tus know what you decide and how it went.