We just flew in August with our 2 year old. This trip she brought her own small backpack with small toys and snacks. She also enjoyed playing on her Daddy's ipad and itouch. I tried to pick up new toys for her backpack that she had never seen.
In my bag, I had a change of clothes for her, diapers, wipes, an empty sippy cup, and a small blanket. I found that it was nice to have some pull up style diapers. We didn't have much time between our connecting flights. She had a skirt on, so I was able to slip her wet diaper off, and pull clean pants on and no one could see anything.
When it was time to board, my husband would take the stroller, and carseat to check at the gate and the bags that he could and he boarded first. I waited with my DD, so we could be last to board the plane. This way we could go to the bathroom at the last moment, and walk around. The less time on the plane, the better!
I made sure to pack some of her favorite snacks that I could give her for take off, so she would be chewing.
I hope this helps!