My birthday is Christmas Eve and it's never been a problem for me!
When I was little, my parents made sure that all the Christmas preparations were done by 23rd Dec, so the celebrations could start on 24th Dec and they weren't using Christmas Eve as last minute panic rush buying.
They also used to give me a kind of "half birthday" in the summer - no party or anything, but I would sometimes get an outdoor toy like a swingball set or whatever that would not have worked in the middle of winter.
I often used to get a trip to the pantomime for my birthday and the panto stars would read out my birthday along with any others in the audience. And when I had bigger (whole-class or similar) birthday parties my friends' parents were always really glad to get their kids out of the house for a few hours on Christmas Eve :-)
A few people used to give me "joint presents" but it never bothered me. My parents always made sure I had separate presents from them, birthday presents wrapped in birthday paper and Christmas wrapped in Christmas paper; and I always had a separate birthday cake not just "Oh, we've got lots of Christmas cake already so we won't bother with a separate one".
Long story short - I think kids like their birthday to be special for them and it's perfectly possible to do that on Christmas Eve, just maybe takes a tiny bit more forward planning!
[Aside, my daughter was born on New Years Eve and I think between us we've got the best two birthday dates in the whole year! She'll always have a party on her birthday! Much better to have a birthday right before a big general celebration when everyone is in celebratory anticipatory mood, rather than right after when everyone is hung over and fed up and on the down slope after the big bash!]