Hmmm - Princes St Gardens is tricky, because on the one side you have Princes St, which has a dearth of eating places and on the other, a bloody great hill!

There is a cluster of restaurants towards the top of the Royal Mile - the aforementioned Hanams is one. I don't know how fussy your toddler is, if they must have dinosaur shapes or if they are happy to try foods, but given how good the staff are in Hanams, that could be an option. Filling Station is a semi-chain - in that it has a handful of restaurants throughout Scotland. It's not amazing dining, but it is a solid American type diner with pretty unexceptional food which will likely cater happily for kids. In the middle section of the Mile, it's probably your best bet (the bit around St Giles is a section where the restaurants can be a bit tourist trappy, and you only eat in them during the Festival when you are desperate for nourishment between shows)
http://www.filling-stations.co.uk

George IVth bridge has a better range of restaurants, including another chain, Pizza Express, as well as a good fish restaurant Ondine, and another Italian doing wood fired pizzas whose name escapes me.

I'm also think that maybe tapas might be good with toddlers - in which case Cafe Andaluz on George St would be good.

And for a decent Scottish restaurant, stray a little further and try Room in the Town - it's a very relaxed restaurant, so should be happy with kids, serving good Scottish food.

I'm sure there are some obvious ones I am forgetting - or being childless, someone else can maybe help with some more child friendly options!