My friend's son has "half birthdays" because he was born on Christmas Eve. We went to his 2 1/2 birthday party which was a lot of fun....a pool party and backyard bbq. It went well. His cake had 3 candles, but the third candle was pushed in deeper to make it 2 1/2 candles, which was really cute.
My husband's birthday is 12/22 and he told me years ago when we were just dating that he hated his birthday because noone ever remembered, or it was just kinda stuck on with Christmas, or he only got one "joint" gift instead of 2 gifts. (Happy birthday and Christmas, and give him a book or record, lol). One year he got a bicycle tire for his birthday and had to wait 3 days to get the rest of the bike. That's funny, but mean. Also, he likes summer. For his 50th birthday, we had a huge bash that was all island themed--far away from Christmas stuff. That was awesome, and a surprise party for him b/c he certainly wasn't expecting his birthday on June 22. HOWEVER that was just that one year. Usually what we do is make 12/22 all about him, and more an EVENT than gifts. We do some special event that may be a little seasonal, but for him. (Stuff in the past: hockey game and a "guy party" at our house afterwards; one year I got a beautiful cottage in the French Quarter and I cooked an amazing dinner, and then we hit the town; one year a bunch of us rented a limo and went looking at Christmas lights and ended up at a friends house for a laid back little party, etc). Then Christmas is "Christmas" and his birthday has rocked his world already.
As for parties: my bff's son has a birthday on 12/27 and what she figured out for him---noone came to his party the first couple times. So then we got the idea to hold his party the weekend BEFORE so it was before Christmas and people were super busy. Then when sending out invitations BEFORE school let out for the holidays, she kind of marketed the party as a "drop your kid off for ice skating at __'s party and have a few hours to shop, drop, or handle the last minute details". Funnily, all but 1 kid who was out of town showed up for that party, and they have ever since. There's a few options for ya....hope one helps. If it were my child, I may do a "friend party" the weekend before, but I would have a special dinner of HER choice on 12/24 and save the Christmas dinner and stuff for 12/25.
I wouldn't want to do a January party because all 4 of us, and 4 friends, have birthdays between October 17 and December 22. Plus we love Thanksgiving. And Christmas and the 1-3 projects we do every year for the holidays, and attending other peoples' Christmas parties too, and then there's New Year's Eve and then our anniversary on Jan 2. So by January 3, we are partied out and won't do another thing until Valentines day. January is our month to get back to the grind of "normal life" and try to recuperate. I'm also of the opinion that if someone is a good enough friend to come to your party, then you should have their number (or parent's number) and email address in time for the holidays be it winter or summer breaks.