I found a lot of tips for traveling with toddlers and a lot of tips for traveling with babies/newborns, but not a lot for traveling with a one-year-old. They are their own beast. They aren't babies who like to sit still and sleep a lot yet they aren't toddlers who can be bribed and like screens.
A lot of traveling with one-year-olds is luck. Your best bet is to stack all the odds in your favor so that your trip is as painless as possible for you and your fellow travelers.
Of course I am no expert, however we have taken our baby on 10 flights so far and he is 14 months old. Here are the best tips I gleaned from friends and blogs.
1) This age generally does not like screens. If they do, only for short bursts of time. We found a way around this by downloading some action/reaction games. Think iPhone and iPad games where you can draw. This was the only screen thing that worked for us. We also try to save screens for desperate times due to their short attention spans. We usually only turn on the screen for landing, when babies are most likely to lose it.
2) Dress them in their cutest outfit for the flight. It's harder to dislike a well-dressed baby.
3) If they do have a major meltdown on the plane, take refuge in the bathroom. Those walls muffle sound really well. Think how loud it is in the bathroom when you flush the toilet, versus how loud it is outside the bathroom. People will still be able to hear your screaming baby, but it won't be ear-splitting.
4) Go with the flow. Our kid is not big on toys. He was amused by the things around him, such as the seat tables that opened and closed, the seatbelts, beating on the in-flight TV, the barf bags etc. Let your one-year-old play with whatever piques his interest.
5) If you do decide to bring toys, bring office supplies or toys from the dollar store. Think stickers, post-its, notebooks for ripping, straws, things that open and close, small things that can be put in containers (barf bag?) and taken out again. Don't let your kid see them before you get on the plane to increase the novelty. One blog I read suggested wrapping them in crepe paper for the kid to unravel. That didn't really work for us.
6) Everyone knows the trick about breastfeeding or bottle feeding as you take off/land. Sometimes that doesn't work out though. Some other things that work are: sucking on a pacifier, crunchy snacks, and baby food in packets. Most one-year-olds aren't all that interested in liquid feedings, but anything that gets their jaw moving will appease their tiny ears. Start with the food right as the flight attendant announces the descent, or sooner if possible. Ask the flight attendant to warn you when you will start losing altitude.
7) Try to sit near other kids. This age loves kids about the same age or a little older. Extra bonus if you find a seat next to a girl aged 8-12. They will usually be as interested in the baby as the baby is in them.
8) Ask for a bassinet. Some airlines carry bassinets, especially on long-haul flights. They say they will only give them to 9-month-olds or younger, but most flight attendants take pity on parents and hand out bassinets willy-nilly. They don't REALLY fit in there, but it gives you a place to store your stuff. And if they do fall asleep and you're able to transfer them, it suddenly becomes the easiest flight ever.
9) A little infant Tylenol and infant Benadryl go a long way. Of course follow the recommendation of your doctor and use weight-based dosing scales, but I find it helps everyone feel at ease to have these two bad boys in your carry-on for sleep/teething emergencies.
10) Try to make sure they get a good night's sleep and their normal naps beforehand and stick to their schedule as much as possible. Most of the meltdowns happen when they are overtired.
11) SNACKS! Our one-year-old is the slowest eater. We bought ourselves so much time giving him snacks.
12) Eat your in-flight meal in shifts if there are two parents there. I had the stewardess hold mine in the back and I went to get it hours later. It's too hard to both eat at the same time, don't bother trying.
13) Take full advantage of gate-checking the stroller. We didn't really end up using the stroller, but you need a lot of stuff when you travel with children. We had the baby in the backpack and used the stroller to tote our suitcases through the airport.
14) Look online ahead of time on the airline's website to see what is allowed for children. Most lap-children are allowed their own piece of checked luggage.
15) If it really is a disaster just repeat to yourself THIS FLIGHT WILL END over and over again. But, if you go in prepared, it probably won't be a disaster! Good luck!