When is the best time to book when flying with young children? Midday or Redeye?

My first instinct is to book a redeye flight, but I'm worried that it will backfire and my kids will be too excited to sleep, and instead keep everyone else up in the middle of the night!

My grandma (children's great grandma that they have never met) had open heart surgery last month, so we feel like we need to go see them this year (they're in Taiwan) before it's too late. My grandparents are both alive, but getting so old! We are thinking about going sometime in the fall.

Any other tips for flying with a 4 and 2 year old? Not worried at all about my 4 year old. More worried about the 2 year old!! And he is definitely in the throes of his twos...