Cheap cuisine, with pretensions
One example of this is Haggis, the famously disturbing Scottish sausage. Composed of the heart, liver, and "lights" of a sheep, the dish is essentially ground up chunks of unattractive internal organs that are then mixed with fat, oatmeal, salt, and pepper, sewed up in a stomach, and cooked. It usually tastes like peppery ground beef, although some cooks manage to give it a more liver-y overtone. Although designed to make use of the less popular parts of a sheep, it has somehow become something of a gourmet dish.
While haggis is a somewhat outrageous outlier, it is only one of many sausages that are made from bizarre portions of animals. Even common salami is composed of meat of unknown origin, while the etymology of head cheese is almost disturbingly clear. American bologna doesn't seem to have any limitations upon the origins of its meats, while mortadella is best known for its huge chunks of straight lard. Black pudding gets its dark coloring from clotted blood, although its flavor is dominated by spices. Blood pudding, on the other hand, has a coppery taste that clearly delineates its origins.
In this context, the idea of artisinal or gourmet sausages seems almost laughable. After all, the original concept behind this cooking method was to simultaneously preserve meat while also making it possible to ignore what one was eating. The combination of chunks of fat and intense seasonings hid the often questionable nature of the meats themselves.
Of course, sausages are only one example of an organ-based cuisine that has somehow gained prominence despite its plebian origins. The most obvious of these is probably foie gras. Basically the liver of a duck or goose that has been gorged on grain, foie gras is creamy, fattening, and almost obscenely expensive. It is also, surprisingly, the result of necessity. Although originally developed by the ancient Egyptians, foie gras became popular among medieval Jews, for whom ducks and geese were a key source of kosher fat. With the renaissance, European nobility discovered the wonders of creamy liver paste and transformed the Jewish survival food into a popular (and extremely expensive) delicacy.
Even today, with pricey internal organ meats dominating the haute cuisine demimonde, it's worth noting that some of their more down-market cousins are still easily -- and cheaply -- available. Chicken livers, for example, tend to be very inexpensive, and can be used to make an amazingly delicious pate. My personal favorite recipe is this one from Epicurious, but there are numerous possibilities out there, from this more traditional version of Jewish chopped liver to this one which is made from mushrooms.
And, if worse comes to worse, there's always the headcheese...