I'd have thought the National Gallery caff I mentioned up the page would fit, being in the gardens themselves, and being an adequate mix of reasonable food and what the USA euphemistically calls "family dining", plus from many of the tables there's a great view out over the gardens and the Old Town.

One thing to point out to first-timers is that, while a number of restaurants etc will be child-friendly, Edinburgh overall isn't, especially if you are using a pram/stroller. The very considerable hills in the centre, myriad footpaths with steps, and streets with cobblestone surfaces, must make wheeling round a problem. It's a steep enough climb on foot from Princes Street up The Mound and Ramsay Lane to the Castle; doing it while pushing an infant is going to be quite a puff.