Nothing that is too small or otherwise easily thrown! When I flew from Corpus Christi to Chicago with my then-14-month-old, I had just made him some really neat toys: baby food jar lids with pictures of family members pasted inside (since we're currently stationed nowhere near any family). Imagine my total mortification when he started treating them like frisbees--they were flying EVERYWHERE! Fortunately, all the nearby passengers found it amusing and were very helpful in corraling the runaway lids. I put those away for later.
Just a quick comment: I don't agree with the statement "It's not your mess" when it comes to messy toys on planes. Sure, you may not feel the need to clean it up (although I would, if my children made it), but someone will have to. It's much better to be on the flight attendents' GOOD side by being thoughtful of them. They'll certainly be much more inclined to give you a hand (bring you extra drinks, etc.) if you're not making their job harder than it needs to be by letting your child smear play doh or anything else all over the place. That's not to say you can't bring play doh, but I'd bring plastic wrap to wrap the tray table (you can just ball it up afterwards!) and supervise her carefully.
Also, with the flight attendents, if they don't offer something to your child for her flight, ask if they have something for children. Southwest Airlines always gives my son something (we've flown a lot!), whether it be a coloring book and crayons, or a trading card with a picture of the plane on it, etc. No matter what cool toys we've brought, they definitely take second place to something he got ON THE PLANE. Apparently that takes the cool factor WAY up, even to a toddler!
Oh, and I also agree with the child having his/her own seat, regardless of age. On that Corpus Christi-Chicago flight, we had a connecting flight in Houston and ended up sitting on the tarmac for several hours (due to a stopage in Chicago). Since I was flying solo with my son I hadn't wanted to try and lug his car seat and decided to only buy one seat and have him sit on my lap. That got very old VERY fast! And also plan on the possibility of a delay of up to a day or more, and make sure you have enough diapers, juice, food, etc. Baby food and juices are SUPPOSED to be exempted from the liquid limitations at security, but not all security personnel know this--in fact at one airport the security personnel downstairs said we couldn't take my son's baby food on, but the ones upstairs said no problem!--so check out the FAA website and print out the relative rules right from there. Then, if security gives you a hard time, you can show them the print out; if they still tell you no, ask politely--hard as it will be in the face of ignorance!--if you can speak to a security supervisor. But that's a worst case scenario--I've never actually been not allowed to take baby food and juices through security.
Have a great trip!