1. Do not buy your toddler or preschooler or elementary aged child a handheld video game.
2. Teach your child to wait in doctor waiting rooms and car service waiting rooms et cetera by reading books rather than using electronic gadgets. Parents model this for your children.
Do not use noisy toys in waiting rooms lest it bother others.
Do not use MP3 players for music, keep their ears open. You want them present in the moment and tuned in to the real world around you. Talk to your kids about what is happening in the place you are in, in quiet voices for some of the time. Let them hear what is going on where they are.
Letting them hear silence in a quiet doctor's waiting room and helping them learn to be quiet with their thoughts is a good thing.
Reading a book is a good way to balance being quiet with putting attention to the book. They can choose when to immerse into the book versus when they want to look around and see what is happening where they are.
My younger son age 3 years and 3 months waiting in a doctor's waiting room, reading a book. He was already starting to read at that age - he taught himself at first or learned naturally by overhearing me give phonics lessons to his five and six year old brother.