To be honest, I struggle to find a decent cup out and about. De Melo on Linthorpe Road aren't too bad if you like your beans dark - Imran who runs it has a blend roasted for him and it's roasted til just after 2nd crack. It's drinkable without sugar. He does ice-cream, desserts, shakes, waffles etc so he expects kids but there's nothing for them to do except sit.
Someone mentioned a place in Yarm, like a micro cafe but I've not been yet. The Baker Street Cafe and Chilli Cake Deli seem to be well liked but again I've not sampled their coffee.
My daughter does horseriding in Saltburn and I take her to a cafe afterwards to warm up, called the Sitting Room. Tiny place, very quirky but they do a half decent French press with beans from a local roaster in North Yorks.
Rounton Coffee from North Yorks often visit the specialist markets in Stockton.
My guilty pleasure is going to York purely to have a cuppa by Gordon at The Attic/Harlequins.