I found an interesting page that gives five different explanations as to where the cheeseburger originated from. You can decide for yourself which story you would like to believe.
Before we even start talking about cheese we must trace the history of the plain old Hamburger. This story goes as far back as the 11th Century when the Mongols carried flat patties of meat with them on long horseback trips. These Mongols would travel to Moscow and the idea of a flat patty of meat spread through the city and was eventually brought to Hamburg in Germany by sailers. From here it spread to the New York and the meat patty became known as a Hamburg Steak or Hamburger.
Tracing the source of the Hamburger (the meat patty in two pieces of bread) gets a little more tricky. The actual hamburger bun is said to be invented in 1916 by Walter Anderson, a short-order cook and founder of the first Hamburger chain â€˜White Castleâ€™. But the Hamburger was around much earlier and was made using two pieces of normal sliced bread.
One claim of inventing the Hamburger sandwich comes from Charlie Nagreen of Seymour, Wisconsin, U.S.. In 1885, he tried selling fried meatballs at a county fair, but customers found them hard to eat while walking around the fair, so Nagreen flattened it and made it into a sandwich he called the “hamburger”.
Hamburg, New York, U.S. also claims credit for the invention of the hamburger. This village celebrates a “Burgerfest” every summer, held to mark the anniversary of the hamburger’s creation at the Erie County Fair in 1885 by the Menches brothers.
Another claim is made by a small diner in the town of New Haven, Connecticut, U.S., named Louis’ Lunch. It is sometimes credited with having invented this quick businessman’s meal for busy office workers in 1900. Similar claims are made from almost every US state!
The first cheeseburger was created between 1924 and 1926 by a chef named Lionel Sternberger in Pasadena, California, USA. The anecdote goes along the lines of a passing homeless man suggested Sternberger should add a slice of cheese to his hamburger order, Sternberger then added this to his main menu and the cheeseburger was born.
Other places have claimed the invention of the cheeseburger as part of their local legend. Louisville, Kentucky-based Kaelin’s Restaurant has claimed to invent the cheeseburger in 1934.  The following year, the trademark for the name “cheeseburger” was awarded to Louis Ballast of the Humpty Dumpty Drive-In in Denver, Colorado.
So we can’t be 100% sure who first had the idea of slapping a piece of cheese on top of the burger patty to create our beloved cheeseburger, but we do know that the cheeseburger is an all American creation that has been exported all over the world. We love the cheeseburger.