Thanks to all those who responded with suggestions for breakfast in the LB area. As we needed something early and omelette centered, we went venturing for ourselves, and stumbled upon Babette's Feast (which I believe is in the Belmont Shore area). Very nice -- good omelettes, fresh ingredients, good fruit salad.
My reservations are about the place is that it's the type of place where you stand in line to order and they bring the food to you; my feelings run along the lines of a (mis)quote from Gosford Park...
"In America they serve you at breakfast."
(Also, refills on coffee are not free, which is a downer at breakfast.)
4621 East Second Street
Long Beach, CA 90803