During residency, what are/were your favorite ways to spend time with your children in the 3-5 hours/week when you were not working or sleeping?
Playing with them. On the floor, TV off, giving them my undivided attention.
That being said there are a bunch of my son's baby pictures that include a William's Textbook laying beside him! I think the key is to maximize the amount of time you can spend with them at home. Also, if possible, to have your spouse bring them up for visits when you are on-call.
I planned to have a child when I was a resident, so I decided early on try to learn everything I could in my first 2 years of residency. My theory was to learn everything in 2 years, so I wouldn't have to study as much when I had a baby. I constant carried notes, photocopies of text book chapters, and notecards with me. All those little moments of down time: waiting for a case to start in the OR; waiting on L&D for a patient to deliver; waiting for your attending to arrive for rounds, these were all my extra study times. And I didn't even have an iphone back then!
This allowed me to rock as a resident and also take time management to a new level. Of course I didn't learn everything in my first 2 years, but I did learn a lot more some of the other residents and I didn't have to study as much at home. Which gave me more time with my souse AND my son when he was born.