You've got a problem. Tucumcari is a dying town and has nothing but a few chains and some very mediocre local places. I think your easiest bet is to continue westward on I-40 for about an hour and eat in Santa Rosa. I recommend Joseph's Bar and Grill at 865 Hwy 66 (ph. 505.472.3361) -- great burgers and New Mexican plates, excellent breakfasts. I used to eat there all the time when I was going to college in Portales and driving to Albuquerque for job interviews.
That's easiest. What you should
do is this: drive about 80 miles south of Tucumcari to Clovis. Eat at Poor Boy's Steak House (2115 N. Prince Street, ph. 505.763.5222), which is the best steak house in eastern New Mexico and has decent vegetarian options. Or if you're more in the mood for diner food, the Foxy Drive-In (720 W. 7th Street, ph. 505.763.7995) has my vote for best drive-in in the country. (They have inside tables as well, where you call the order in from a red phone next to the napkins.) Plus, for extra authenticy, it's directly across the street from Norman Petty Studios, a landmark of '50s rock and roll!
From Clovis, you take US-84 to Fort Sumner and then Santa Rosa, when you merge onto I-40. About 15 miles later, you get off of I-40 and back onto US-84 North, which takes you straight to Las Vegas. It's a two and a half to three hour drive. I just hate the idea of someone eating in Tucumcari (shudder) when there's Poor Boy's and the Foxy just over an hour away.
Where are you going to Tucumcari from? Unless you have a desperate need to go to Tucumcari for some particular reason, I assuming you'll be coming from towards Amarillo, in which case taking US-60 west out of Amarillo instead of I-40 west will get you directly to Clovis in less than 100 miles (not much longer than the trip from Amarillo to Tucumcari), which puts Clovis not very far out of your way at all.