I hope this reply is not too late. For Arizona Mexican I would suggest Romo's, El Rancho or the Wayside. For breakfast or lunch I like Doughnuts plus. They make great pressed sandwiches.
HI! I live in Holbrook. I wouldn't say the food scene is amazing but there are several local places I love. First for breakfast/lunch there is a doughnut shop called Doughnuts Plus on Navajo Blvd (One of the main streets).
If you are going to be going through Heber around lunch or dinner the Red Onion does great burgers. It's on the left after circle K as you are driving towards the valley.
If you are wanting something more upscale/foodie but also more expensive check out the Turquoise Room in Winslow. It's a 30 minute drive from Holbrook, west on I-40 and it's in an old railroad Hotel the La Posada. http://www.laposada.org/