Hi, I thought the Kleenex idea was great. Another thing my son loves to do is to pull out dental floss in one unending string and unroll toilet paper.
I also really liked Nancy L's ideas.
I'm not a frequent flyer, but I brought my son to Arkansas to see family when he was a year old. I held him on my lap the whole time.
Things that worked to keep him busy and quiet:
*nursing
*sleeping
*other passengers wanted to play with him and hold him, so other people held him and he tried to grab fingers they poked at him through the seats :D
*he LOVES snacks. I often mix up several kinds of cereals in a Ziploc bag. He loves to pull them out and gets a different kind each time.
*it turned out that he wanted to play with the ice in my cup of water
*after the ice and water were gone he had a GREAT time putting cereal in the cup and taking it out again -- for a LONG time.
*using toys in new ways, like wearing his squid rattle as a hat -- made him laugh endlessly
*I got frustrated with his wiggling and turned him over to play him like a guitar. The other passengers thought I was weird, but he laughed.
*pop goes the weasel. He loves the song. I trace a circle on his tummy or part of his body as I sing. Then I gasp and sing, "POP goes the weasel!" and give him a little lift or tickle. He goes from screaming to laughing within seconds.
Now he's two and we went on a car trip of 5 days to Arkansas. Some things that worked, which might work for a one year old:
*laptop with a Baby Noah DVD
*lots of dollar store toys, doled out slowly. Be sure there's no lead paint in case he chews on them. My son loved cars and those plastic tube things that are squishy and full of coloured liquid. It says not to give to children under three, but I didn't let him play with it unless I was watching him the whole time.
*when it was dark he loved glow in the dark sticks
*he played with my hair A LOT :D
*slowly dole out snacks -- have a variety if you can
*on the plane ride home he discovered a 10 month old baby in the seat in front of us. The dad was very understanding about having his seat back kicked despite my best efforts at curbing it, and Nathaniel occupied their baby so that it quit screaming. They kept smiling at each other and holding hands.
*if you get a chance, let the kid be active to run off some energy (like in the airport).
Good luck!