Whist not a restaurant as such, Sunday Walking Street (Sunday evening street market) is a great place to go to sample local street food and delicacies, and the prices are dirt cheap too!
It's a lot of fun browsing the hundreds (may be thousands?) of stalls selling all manners of goods and handicrafts, often handmade by the person selling themselves, and to sample some of Chiang Mai's local food as you go.
If you need a "proper" meal, you can always drop into one of the temples on the main street of the market. There, you'll not only find the buddha, but food nirvana as well - Ok, I might be exaggerating a little. I mean, you'll be able to find hundreds of stalls selling food! Think of food court in a temple. Cost starts from 15 baht or so for a small Pad Thai, so why not try a few new things?
Here are some of the food found on Sunday Walking Street:
Deep fried tofu with sweet peanut sauce.
Gyoza - deep fried dumplings
I forgot the name of this one, but it's like a type of rice and pork salad. Very nice, actually.
This is the stall selling oyster and various seafood omelette. Looks great, but unfortunately, the omelette itself wasn't. I've had better. It was fun trying though!
Other things to try - "Ancient Ice-cream" - they are little blocks of ice-cream in various "exotic" flavours, or you can go for coconut. You can't go wrong at 10 baht per piece. Go on, try something different!
On every Sunday evenings, Rachadamnoen Rd, starting from Thapae Gate.