DD's 1st birthday was 26th December. (As have every other birthday since, to be fair!
).
As she was also the first grandchild on both sides, we actually held back a good bit.
I bought a copy of "Twas the night before Christmas" for Christmas Eve bedtime story - and she still insists on hearing that for the first time every year on Christmas Eve (she's about to turn 12).
We got her a few toys - a small round activity toy that lit up and made noise, a shape sorter and a tractor/trailer/cow and horse set. Her stocking had some chocolate, a second set of toddler cutlery (to reduce pressure on washing up) and a spare drinking cup.
Under the tree, we had an outfit wrapped from us - we always have a present from Mum and Dad under the tree, the same as we both grew up with so it doesn't all change hugely when Santa no longer comes (we both were the eldest of a number of siblings).
For her birthday, we got her 2 more books (one stories for bedtime, and 1 for her reading/playing with in daytime) and a jigsaw.
Also, she had the traditional first birthday cake in our family - a jelly, made in a pudding bowl and turned out, with the Child's name piped on with whipped cream. As not all small DCs like cake, but I've never yet met a DC who doesn't like jelly - and all small DCs should be able to enjoy their own first birthday cake!!
We do go for more traditional cake for 2nd and later birthdays, we're not complete loopers!