Only cereal was specifically invented as a breakfast food. According to Heather Arndt Anderson's book Breakfast: A History, there was a morally driven health crusade in the late 19th century in the US. Their broadly prescribed idea was abstain from meat, alcohol, tobacco etc. Many "sanitariums" were opened that supposed to improve the health of clients and each place formulated their own menus based on this diet. A prominent figure of the movement was Sylvester Graham who advocated a shift to simple grain-based breakfast and was particularly obsessed with unprocessed flour (and invented graham flour). Another man named James Caleb Jackson who opened Our Home on the Hillside Health Spa tried to create an easy breakfast by first making hard wafers of graham flour that had to be soaked in milk overnight (which defeats the purpose) and he later improved it by breaking the wafer into tiny granules that resulted in something similar to Grape Nuts that was advertised as "ready for immediate table use". Around the same time owner of Battle Creek Sanitarium John Harvey Kellogg invented rolled flake cereal (supposedly he tried to salvage cooked wheat that has been left on the stove overnight). They patented the product in 1895.
Various forms of pancakes have been eaten for millennia. According to Pancake: A Global History by Ken Albala, one of the earliest forms for pancakes familiar to Americans is from a 16th century book called Goodhuswifes Handmaide for the Kitchin, however most recipes today were derived from Hannah Glasse's 1747 cookbook The Art of Cookery. Cornmeal pancakes called johnny cakes or hoecaks were often made by freed black cooks around the same time. Reportedly George Washington enjoyed cornmeal pancakes "swimming in butter and honey". By the mid-19th century buttermilk pancake became commonly referred to by cookbooks as a breakfast food.
I'm not sure if muffins are as associated with breakfast as pancakes and cereal (OP should probably specify muffins are considered breakfast food in the US). Most early American recipes for muffins were similar to what Americans would call an English muffin and they were something served for tea time instead of breakfast (and still the case in the British Isles). On the other hand these cookbooks often contain recipes for tea cakes in the same tea time recipes that are much closer to American muffins. According to the Dictionary of American Regional English, the first time American muffins were called "muffins* was cited is from a 1879 book called Housekeeping in Olde Virginia: "Another recipe for muffins...make the batter the consistency of pound cake, and bake in snow-ball cups as soon as made."
Aside from being specifically invented for breakfast like in the case of cereal, most breakfast food were obviously popular for convenience. Considering that most food made in the western world used to require a long and slow cooking that's impractical for morning, so food that could be cooked quickly quickly or could be made ahead of time became popular as breakfast food.
The modern concept of "breakfast food" was the result of a public relations campaign by Sigmund Freud's nephew