Also for in the car, audio books are great. I am pretty sure that much of my kids vocab was from such gems.
We always welcomed the gift of an Annual Family Pass to the zoo, children’s museum, or all ages museums.
With a family pass, we would go more often, but we didn’t feel the pressure to stay if the kids were having a bad day for whatever reason.
So, so, so many fond memories of our excursions to Chicago or Milwaukee.
YES to the cardboard boxes. Both my kids crawled through large packing boxes, shredded wrapping paper, played with tin foil, etc. and had a blast. We told folks just to bring paper and boxes that could stack or make noise.
My kids were really fond of old Disney musicals between ages 2 and 5 – the kind with live actors with British accents and lots of rich vocab! Ditto Beatrix Potter and the ORIGINAL Thomas the Tank Engine episodes and book.
One year old really doesn’t need/want that much. I think contribution to 529 or cash probably be very much appreciated by the parents. We lived in NYC when kids were young, I dreaded large and noisy toys for my kids.
There’s something called a Gertie Ball. It’s soft and can be easily grasped with baby/toddler hands. My kids played with it for years. It’s perfect for babies and fun for everyone else. You can get it at Target or Amazon for about $10. https://www.amazon.com/s/ref=nb_sb_noss_1?url=search-alias%3Dtoys-and-games&field-keywords=gertie+ball
Toddler flip open chair or couch https://www.amazon.com/Marshmallow-Furniture-Minnies-Bow-Tique-Flip/dp/B00EVQJ2NK/ref=sr_1_1?ie=UTF8&qid=1487032526&sr=8-1&keywords=toddler+couch
Various bath books/toys
And this, my favorite gift for 1 year old bday present: https://www.amazon.com/Fisher-Price-Laugh-Learn-Around-Learning/dp/B01ASVCTIG/ref=sr_1_2?ie=UTF8&qid=1487032658&sr=8-2&keywords=fisher+price+standing+toy
even though the mom doesn’t like plastic blinging things, the kid will like this and it isn’t obnoxious.
recorded books -can listen in the car. Good for development. Big baby bibs, sippy cup, divided plate, and fork /spoon
Everyone above has taken care of the book idea - except for, if you feel baby has a lot of books already you can buy some books that might be enjoyed in a year or two - board books till at least age 2 is what is developmentally appropriate.
But for a different idea (unless I missed it above) is a push or pull toy - you know, like the old push “popcorn popper” toy or the little Fisher Price dog on a leash that you pull…
Think about what baby’s next developmental milestone is/will be - first words, pointing, walking - and choose from there!
A dish towel, a box, something to grab and bang like a stick. Or, buy stuff to impress the parent because the kid won’t care or notice.
How about clothes,diapers ( if they use disposable) or Series E bonds?
If it is a child you will continue to give gifts to over the years, you might get something like a wooden train that you can give more pieces to until he/she builds up a big set, like a Brio or Thomas the Tank Engine.
A little slide is pretty fun at age one. My daughter had some OT/PT needs and went up and down hers about 10 bazillion times. I gave my twin nephews a Playhut set when they were about 15 months old. I was looking for a tunnel made out of parachute material and found a whole set with tunnels and houses for just a little more. They loved them and when they were 3 or 4 put them over their beds to make caves to sleep in.
@Dustyfeathers you listed just about all the books I would have! Especially -* Freight Train, Go Away Big Green Monster* and “anything by David Wiesner”. I’d add Vera B Williams’ * More, more more said the Baby * to your list. I think one, is probably still a little too young for duplos - they’re still sticking stuff in their mouths. My oldest really liked this Tupperware shape sorting toy at this age: http://www.sears.com/tupperware-shape-o-ball-toy-red-blue/p-SPM3708512521?sid=IDx20110310x00001i&gclid=CjwKEAiAz4XFBRCW87vj6-28uFMSJAAHeGZbVOQ64IgKWrKQWHbjbMQAw8cUXNeLDZ2Mazxxw9kedxoCBoDw_wcB&gclsrc=aw.ds
Best gift for a one-year-old? A hug and a “happy birthday, cutie pie.” S/he doesn’t know it’s her/his birthday, so why start the madness if you don’t have to? If you’re invited to a one-year-old’s party, you can politely decline in favor of your colonoscopy appointment. Nothing good can come from attending a party where the guest of honor is not in control of bodily functions.
Yes, I am a Scrooge. Our own kid didn’t know about birthdays until he was four, and we certainly didn’t subject other adults to them.
If you LIKE these things however (or you are a grandparent), there are some good suggestions above. Carry on. 
I agree that what the gift is matters little to a one year old, but I really enjoyed the parties – gives you a chance to spend some time with the child. They are adorable at that age - sweet and innocent.
The most celebrated gift I ever brought to a one year old’s party was the huge, yellow latex “smiley face” balloon that was tied to the gift. All the kids went crazy for it and chased each around the house to get hold of it. The mom had to confiscate it and put it in a closet to stop the chase. It’s the thing you least expect that gets their attention.
Because of my username, I feel I have to suggest getting a puppy 
Ditto on the books!Great habit to get into is reading to young kids. Eventually you will need the public library as there isn’t room to store all of the variety one should read. I learned a lot about all sorts of things reading to my kid.
One year olds are easy to buy for as toys are their learning devices.
NO! to a puppy. The child will not be the caretaker and the parents need to be the ones making that decision, at any age. Dog lover here but it is such an individual, forever, decision.
My D and I put together a “book of the month” gift box for my nephew when he turned one. My D and I are voracious readers, and wanted to pass the love of books. So on the first birthday we chose a classic that she loved, and wrapped it with a note that each month a new book would arrive. Well, here we are 13 years later, and we have provided a book every single month. My nephew has a love of reading and loves getting the books. I did tell him the “club” ends on his 18th birthday.
Both my kids at that age LOVED small pull wagons – since walking was still a relatively new trick – in which they could pull their “stuff.” Here’s an example.
Also, my daughter in particular was very attached to wooden pull animals from Brio. Cute and very good quality. I saved them for her kids.
https://■■■■■■/nXfw4D
We got little stuffed chairs (one teddy bear, one panda) for about $20 at ToysRUs when they were about a year old. The guys loved them. They watched videos and read books in them until they were 4-5 yo, except when they weren’t sitting in a lap or curled up in bed with one of us. Good memories!
We used old packing boxes to make a fridge and stove. Bought a bunch of Fisher-Price kitchen supplies and play food, and the guys had a BLAST pretending to cook (or crawling into the boxes). I attribute this to S2’s wanting to learn to make mac & cheese and brownies on the real stove when he was three (he got to mix ingredients and pour; we didn’t use the stove til later), and his lifelong love of cooking.
Here’s another I like, https://woodencaterpillar.com/ Very good for smaller gifts especially.