Chile con queso (or chili con queso) is a simple Tex-Mex dish of chili and cheese. President Lyndon B. Johnson published a favorite recipe in 1964.
 
     
Wikipedia: Chile con queso
Chili con queso (also spelled chile con queso) is an appetizer that is served in Tex-Mex restaurants.
 
Chili con queso, (also spelled chile con queso) which is a part of Southwestern cuisine, originated in Texas. Chili con queso is rarely found in Mexican restaurants outside of Texas. Authentic Texas style chili con queso has been found as far east as Atlanta, Georgia and as far west as Phoenix, Arizona. La Posta, a Mexican restaurant in La Mesilla, New Mexico, near Las Cruces, serves a New Mexican version of the dish. If it is found in another state, the restaurant is usually a part of a Texas restaurant chain such as Ninfa’s or On the Border. Authentic chili con queso is rarely found in Mexican restaurants in California. 
 
Cooks.com
CHILE CON QUESO (CHILE WITH CHEESE) 
2 tbsp. vegetable oil
1 c. chopped onion
2 sm. garlic cloves, minced
4 fresh chilies, peeled, chopped or 1 (4 oz.) can chopped green chilies
1 or 2 jalapeno chilies or sm. hot chilies, peeled, chopped
1 (8 oz.) can stewed tomatoes
8 oz. Monterey Jack cheese, shredded (2 c.)
8 oz. Longhorn cheese, shredded (2 c.)
1 c. dairy sour cream

Heat oil in large saucepan. Add onions and garlic. Cook until tender, but not browned. Add chilies and tomatoes with a spoon. Lower heat. Add cheeses and cook until melted. Stir in sour cream. Cook until just heated; do not boil.
Makes 4 cups of dip.
 
Cooks.com
CHILI CON QUESO (MILD) 
1/4 cup butter
1 onion, finely chopped
1/4-1/2 tsp. garlic, minced
1 can mild green chilies, chopped
1 16 oz. can diced tomatoes, draind
1-3 tbsp. evaporated milk
pimiento or paprika

Melt butter; saute onions until translucent. Add garlic, salt, chilies and tomatoes.
Cut up cheese; add evaporated milk until consistency resembles thick gravy.

Cook slowly, adding milk a little at a time. Stir often.

Hot chilies may be added if hotter than mild Con Queso is desired. Pimento paprika or achiote may be added for color.

Serve with corn chips or Tostitos as a dip or cut corn tortillas into 6 preshaped wedges and fry in hot fat until crisp.

Serves 8-12.
 
Google Books
Cuisine, Texas: A Multiethnic Feast
by Joanne Smith
Austin, TX: University of Texas Press
1995
Pg. 25:
How many Tex-Mex dishes does it take to make a botana? I believe you need four, not counting chips: Nachos, Chile con Queso, Guacamole, and Picadillo. Old Hat? Of course, but they still rank high as favorites for an afternoon of TV football or an evening of video film classics. The combination offers a balance of flavors, textures, and colors and goes well with soft drinks, coffee, tea, beer, and wine.
(...)
Ever since the invention of Velveeta cheese and canned Ro-Tel tomatoes, hardly anyone starts from scratch to make Chile con Queso, but it’s really good. Keep it hot in a chafing dish or in a candle-heated serving dish.
     
17 June 1920, Nashua (Iowa) Reporter, pg. 3?, col. 6:
Pepper is the source of Mexican revolutions, according to many people. Hot tamales, chili con carne, which means pepper with meat; chili rellenas, chili con queso, or pepper with cheese; all the other hot dishes that delight the Mexican palate, are supposed to incite these fiery Latin-Indian folk to actions that people of more mien would not contemplate.
 
18 May 1937, Modesto (CA) Bee and News-Herald, “Foreign Cookery Is Studied By Women,” pg. 12?, col. 4:
Mrs. R.  B. Hogancamp, chile con queso,...
 
27 December 1940, Los Angeles Times, pg. A5:
Walt and Sit Candy—the latter in a bright Yuletide red felt chapeau—were raving over the chile con queso.
 
4 November 1947, Chicago Daily Tribune, pg. 25:
Chili con queso made a bigger hit at my table than even the sout-of-the-border barbecued chicken. Chili con queso is a cheese and pepper toast, um-mm, but good!
 
Chili con Queso
1 chili peppers
1 onion
2 cups tomatoes
2 eggs
Salt
1 cup grated yellow cheese
1/2 teaspoon chili powder
1. Chop peppers and onion fine and saute in a little fat. Use a green pepper if chili peppers are not available.
2. Add tomatoes, stir well, and add while stirring, the well beaten eggs.
3. Add salt to taste, cheese and chili powder, stir and cook until cheese is melted and serve hot on toast.
 
1 February 1954, Los Angeles Times, “Recipe of the Week” by Lola Turley, pg. B6:
This modern version of the traditional Spanish chili con queso, was sent to us by Mrs. Lois Marie Young of El Paso, Tex., and is made with green chilis blended with creamy, smooth cheese and a dash of lively tabasco. This tempting dip is a savory change from the usual line of “nibble foods.”
(...)
Chili Con Queso
Use two and one-half cups canned tomatoes and one small can green chilis. Put both in top part of a large double boiler, place directly over flame. Cook slowly until mushy, adding a little water if necessary. Remove from flame and place over boiling water. Add approximately two pounds cheddar cheese, cubed, slowly and cook until the cheese has melted. Flavor with a dash of tabasco. Serve with tostados, crackers, potato chips or anything with which to “dip.”
 
1 October 1964, Washington Post, pg. D1:
LBJ RANCH dishes will include chili con queso, pickled okra, guacamole and chili dip. All are favorites of the President and First Lady.
(...)(Pg. D14—ed.)
Here are the recipes from the White House.
 
LBJ RANCH DISHES
Chili con Queso
1 No. 2 can tomatoes
1 large onion chopped fine
1 bud garlic chopped fine
1 tsp. salt
4 tbsp. chili powder
1 tbsp, powdered comino seed
1 tsp. oregano
1 lb. aged cheddar cheese
Simmer all this except cheese slowly for about 2 hours, covered, stirring often. Uncover, turn heat up high and stir constantly until all fluid is gone and you have a thick paste. This paste can be frozen. If so, set out to thaw a couple of hours ahead allowing for about 30 minutes in a double boiler before serving time. Add to it, in double boiler, 1 pound best aged cheddar cheese cut up in chunks.
 
Cover and let stand over water that is simmering, not boiling, as boiling water tends to make cheese stringy. Stir occasionally to mix well. Taste and add salt if needed.
 
Serve in a chafing dish with large size fritos if desired, but tortillas quartered and fried in deep fat are better.