Ditto on the books. Discovery books are great for that age; pages with doors, flaps, textures, etc. Wooden puzzles are also great gifts as the child can play with them for years.
I second the books and puzzles. I got DS one of those "roller coaster" toys you find at most doctor's offices. The ones with multiple twisted wires with blocks on the them that you push back and forth. I don't even know the real name, but I got ours at Ikea for $8.
Re: what's a good present for a 1 year old's birthday?
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When it come to clothes, odds are you'll be fine to give him/her PJ's in 12-18 moths size
this is our "standard" gift right now, everyones LOVES it, it is from Target, about $20
https://justb-byou.com/toys/lively/parum-pum-pum/
maybe you could throw in a book or outfit too
This is my go to because it's a great toy that can be flattened for easy storage.
https://www.amazon.com/Pacific-Play-Tents-Multi-Tunnel/dp/B00009IMAS/ref=sr_1_1?ie=UTF8&qid=1300300984&sr=8-1