By the time you get around to a "half-birthday" its so far removed from the child's actual birthday that it may not seem that it has anything to do with the day that they are born. Doing things this way could also fool your guests into believing the birthday kid's special day is nowhere near when they thought it was. Imagine growing up believing your best friend's birthday was 6 months after it actually was?
I know I would have hated holding my birthday party MONTHS after my actual birthday.
I like the idea of a New Year's party but, again, you're planning around a holiday. Luckily, most preparations for New Year's Eve are for the night time so you could still hold a daytime event. Or, another idea would be to host a New Year's sleep over. The kids would get to stay up late and the parents might appreciate the free babysitting.
It might make your son not feel as special having his party on another holiday, though, so then the next best thing might be having it early. You could have it late, but then its technically the next year already and it seems to me like it should be counting toward his following birthday age.
This is pretty complicated. Good luck finding something that works for you.