In my country/culture this is not ever done, no one would dream of asking a friend to pick you up that late. You take that liberty only with a family or you just wait until 6am or so in the airport itself so that it's at least bright when your family drives to pick you up. Here it's not safe for women to drive alone, and even otherwise, it's considered massively rude to ask a favour that would disrupt one's sleep, unless it's a dire emergency.
UNLESS, the "driver" friend has a history of picking up people from the airport (like a side job) and/or the friend is paying for the trouble.
However I know things are obv different in the UK, so here's my question for you, OP: how was the request framed to you? Was it "Dear Friend, Is it possible to pick me up at 3am? Would that be okay? Of course I will gladly pay for the petrol costs". Or was it a blase CF style ask?
By the way, YADNU to say no. It's a request for a favour that will involve you driving in the middle of the night, in a non-emergency. No one is obligated to grand such favors, even to besties and family.
Also I am quite speechless that OP's friend thought it was ok to use OP as a cab service - she is not even visiting or staying in OP's home, she will just get picked up at the airport and get dropped at her own place, courtesy of OP's time and petrol money?! LOL.