Wash, peel, and cube the beet. Place beet pieces in a medium saucepan with water and vinegar, bring to a boil. Reduce heat and simmer, covered, for 15 minutes. Remove from heat; set aside to cool completely (do not drain).
Place the eggs in the saucepan with beet and liquid, making sure they are fully immersed. Let stand 10 to 15 minutes. Remove eggs from liquid and slice in half (discard beet and liquid or you can save liquid to use for dying Easter Eggs ). Remove the yolks and put them in a small bowl, setting the whites aside on a plate.
Mash the yolks with the mayonnaise, mustard, bacon, onion and a pinch of salt. Pipe or scoop back into egg white halves. Sprinkle with paprika, cayenne pepper or coarse salt.