Mels on Broadway (greasy spoon) and the Corner Cafe, also on Broadway. CC makes a great stuffed french toast. The Atlantic Grill down by the beach (middletown side of the line, across from the surf shop) is top notch too.
Corner Cafe works well for my wife and I whenever we stay overnight in Newport. I like the savory irish fry up and she likes the sweet creations. Friendly, caring service makes you feel like you were invited to a friends house for breakfast.