If breakfast is not included in the price of your hotel, we recommend Café Bar la Frasca (Plaza de la Rubia 6, exiting Plaza Mayor on Cronista Lecea). The food is average but the bar and outdoor seating area are very nice, and breakfast is cheap (1.80€ for coffee + pastries or small sandwich / 2.20€ with orange juice).
If you don't mind walking a bit, we better recommend Dakota, on Avenida de Fernandez Ladreda at the intersection with los Hermanos Barral street (walk to Plaza del Azoguejo following the avenue). The place itself isn't very pretty from the outside but the omelettes are delicious and they have tables with views of beautiful San Millán church. It will also give you a chance to see the newer part of the city.
