Our kids love ramen and pretty much all ramen places are the same in their eyes. Since ramen shops are pretty much ubiquitous this makes finding somewhere to eat pretty easy. Unlike the kids WE have some places that we prefer more than others- when we find places that stand out I'll post on them here. In the meantime, (on the suggestion of another Kyoto parent) here is a useful phrase to help order for kids: Negi nashi kudasai- no green onions please. (Nashi is a useful word to put after things the kids don't like- some of our kids don't like ketchup: so we use it a lot. "Kechappu nashi kudasai" "Sōsu nashi kudasai" etc.)
Here is an unusual place- Mamezen. It is "Kyoto style" ramen with soy milk broth,with yuba and plum as toppings. We saw it on Kyoto Foodie and gave it a try- very delicious. It worked well because we were the only ones there, but the small space might not work so great with other diners. The couple who run it are super nice and have a young child of their own, so they looked with understanding at the antics of ours. The other downside is that they open late: 7:30 and we usually prefer to eat about 5:30 or 6. Lunch would be great though. They do have a kid's set that was perfect for our 5 year old.