When my son turned one it gave me the chance to see exactly how he interacted with different toys and games. A lot of friends also have children a similar age, so it has been interesting to see what works across the board.
What I noticed is that special interests are yet to develop. They may show an affinity for stuffed animals or certain toys, but it's really during the following year that they get super into trucks or crocodiles or puzzles or whatever it might be. So for next year, your knowledge of the child will make a big difference to the success of your gift. For now, you can just go for generic crowd pleasers. And no need to spend much - they'll always be most interested in a spatula and cardboard box at this age anyway.
Duplo Train
In my opinion, the best gift for a one year old is a Duplo train. It will be engaging for the birthday boy or girl at one, two and three years old, is educational and is great value.
Note that the manufacturer's instructions label this for 18 months plus, so bear that in mind when deciding suitability. I personally found that I was supervising him closely at one anyway so I felt very comfortable with it being a good fit for his age and it grew with my son perfectly - when he was 12 months he loved taking the pieces apart. When he was 18 months, he was fascinated in putting the pieces together. And at 27 months he was creating interesting towers and shapes on each train car with the pieces and getting creative.
There is also a special Disney birthday train option, which is a pricier option for almost the same thing, but the candle is a fun touch.
I've bought both of these and they worked great.
Play Tunnel
If you are going to a birthday party in a park or someone's back garden, this is an ideal first birthday gift because once it's opened, it can add to the fun on the day.
I researched a lot of options when I got this for my son (and subsequently other friends' kids). I like to put it away for some time between uses and get it out for play on a rainy day. It works great because it's interesting for them when they're babies and then it comes into its own as 'houses' and parts of forts in creative play as they get older.
Picture Books
Kids this age are still too young for stories. But they are starting to understand language and becoming interested in the world around them, so books that help to label things can be particularly engaging for a one year old.
My Mum was an English Literature teacher and when she retired from teaching, ran the children's book department of a famous bookstore. So I wasn't surprised when this was one of the books that was not just very engaging to my son when she gave it to him when he turned one, but has had great longevity long after. What I particularly like about it is that the images are photographic and extremely clear, so a baby doesn't need to work out that the cartoon is the same as the actual object. And the words are clustered in tabbed categories, which helps the child understand their context as they get older. Even though my elder son is now able to follow along to far more complex story books, this is still one of his favorites.
Music Books
One of the most successful gifts my son received when he was twelve months old were two button-press music books from his French cousins. They were so successful, in fact, that they kept him entertained for 15-20 minutes at a stretch. Everything else engaged him for a minute or so. Which meant that I went a little crazy and tracked down the U.S. equivalent, then and bought pretty much everything in the series. Trust me, you'll want more than one, and now they have a bundle package of our two favorites in the series.
One con is that the batteries in these Cali books run out quickly - which will happen when they become obsessed with pressing the buttons - so getting an extra pack of batteries is a good idea (store them safely)! I recommended these to my local parents' group and had a ton of positive feedback.
A general watchout here - do not get a music / sound book where the buttons require actually pushing down with any force, as the child will not be able to operate them at this age. Most button-press music books are no good for one year olds for this reason, but these books are the exception because they respond to exceptionally light touch.
Teddy Bear
It's a classic gift for a reason. If you're imagining it as a bedtime buddy, you may want to look for something that has stitched eyes and nothing that can be chewed off and swallowed. For daytime cuddles, this has been our favorite. On the more expensive side but with plush toys you get what you pay for and wow this guy is soft.
For some reason, our bear ended up being called Sausages and my husband bought him a t-shirt with his name on. It could be fun to do something similar with a personalized message like 'Happy birthday, [kid's name]' for the special day.
It has been so much fun to see how the most successful gifts, all of them listed above, were not just big winners when our son was given them on his first birthday but are still so popular well after his second birthday.