I therefore decided to apply some Turtle Island (North American) inspiration to omelettes. The way it used to work was that some time in spring, the first kid would come running back to camp with a nest's worth of random eggs in his shirt -- most often quail, duck, goose, or turkey. These could be cooked whole in hot ash, or cracked into the hollow of a stone, and topped with whatever fresh greens were sprouting at the time.
I used duck eggs, ghee because I didn't care to find the smokepoint of sunflower oil, sea salt, rubbed sage, a splash of grass-fed unhomogenized milk, canned mushrooms, a couple different cheeses, and for my partner's omelette about 1/8 cup of diced onion left over from making yesterday's elkloaf. Omelettes are fantastic for using up whatever leftover vegetables, meat, etc. you happen to have. A good combination is 1-3 seasonings and 2-5 fillings.
Bear in mind that the different tribes had quite diverse diets. There's no single Turtle Island cuisine, any more than there is a European cuisine. Coastal people ate seafood and sea vegetables, inland folks hunted large game or farmed crops, and so on. South American offerings further broaden the options.
So let's take a look at some options for decolonizing your diet, or just exploring indigenous goodies ...
Eggs: Almost all birds lay edible eggs, although the size and quality vary. Anything from Turtle Island is super legit. Duck and goose eggs are big and rich. Quail eggs are tiny; I haven't tried them. Turkey eggs are pretty chickenlike. Chicken eggs work fine if that's what you have. Try to get farm-fresh eggs from pastured hens if you can.
Fat: Among the common animal fats used by tribal people were bison, caribou, bear, salmon, duck, and goose. Bison fat is someone akin to beef fat. In the southwest, javelina is basically like domestic lard. Sunflower, hazelnut, and walnut all yield vegetable oils. Sunflower has nearly no taste, the others taste similar to their nuts. If you like nuts in your omelettes, by all means match the oil. This gives you good options if you are vegetarian or dislike animal fat. Ghee is clarified butter, ideal for cooking because it's so hard to burn. It's probably not historic, but there were mammals, so you never know what somebody might have tried.
Salt: At various times the trading network of Turtle Island spanned all four coasts, making sea salt available. Some tribes mined for mineral salt or produced it in several ways. It can be smoked to add more flavor. Use what you have and like. Table salt will do.
Herbs and Spices: Sage is ubiquitous as a sacred and culinary herb. Bergamot has a musky, lemony flavor you may recognize from Earl Grey tea. Juniper leaves or berries have a resinous, peppery flavor. Cedar is another resinous, woodsy flavor. Staghorn sumac berries are also lemony but belong to the same family as poison ivy, cashew, and mango so be cautious if you're allergic to any of the other three. Wild mustard is zesty. Wild ginger has a warm zesty flavor. Domestic analogs are fine, and you can also use any foreign flavors you like.
Dairy: There is little if any hint of dairy use in the past, but people ate everything, so presumably if they brought down a lactating animal they would take advantage. With the arrival of domestic animals from Europe, some tribes quickly latched onto sheep and/or goats for meat, milk, and/or fiber. I find that a splash of milk or half-n-half makes eggs blend smoother. I'm also a big fan of cheese in omelettes. But if you are dairy-free then a historic diet will offer you many recipes.
Nuts, nut milks, and nut oils: Tribal people gathered acorns, beechnuts, black walnuts, chestnuts, hazelnuts, hickory nuts, pecans, pine nuts, and white walnuts. These were eaten whole, ground into meal, brewed into milky liquids, crushed for oil, and so forth. Some of these make good substitutes for animal products.
Grains and seeds: Some tribes relied on seed crops such as wild rice, little barley, amaranth, quinoa, chia, and sunflower. Some of these work well as the base for a medley filling. Consider wild rice with ground meat or quinoa with mushrooms, plus some chopped vegetables.
Meats: If they could catch it, they ate it, although everyone had preferences. Primary meat sources for tribes included bison, venison, bighorn sheep, small game like rabbits and squirrels, ducks or geese, and salmon. Most of these work great in an omelette. So do domestic analogs of beef, lamb, mutton, and chicken or turkey. However, only a handful of tribes depending primarily on meat -- most got a majority of their calories from plant foods. So if you are vegetarian or just dislike meat in omelettes, you still have lots of options.
Vegetables: Look at a forest with a gatherer's eye and it's like a giant supermarket. Almost everything has some part you can eat, drink, or use to flavor your food. The Three Sisters are corn, squash, and beans. Also consider potatoes, tomatoes, chili peppers, green peppers, carrots, asparagus, cattail, sunchokes, fiddleheads, and mushrooms.
Greens: A great variety of wild greens are edible raw or cooked. Examples include fireweed, dandelion, chickweed, curly dock, wood sorrel, watercress, bull thistle, red clover, sheep sorrel, shepherd's purse, lamb's quarters, and nettles. Most species that can be eaten raw are good chopped and sprinkled over an omelette. Those that need cooking first can be incorporated into the filling. You are not limited to wildcrafting: some of these can be cultivated or purchased at a farmer's market.
Alliums: Somewhere between a spice and a vegetable lies the onion family. Onions, shallots, leeks, ramps, garlic, chives, etc. have both wild and domestic versions. They're good sauteed or raw. If you want to punch up your candied onions, add just a tiny bit of maple syrup or honey as they begin to caramelize. In spring and summer, the flowers have a milder flavor with floral notes and a bit of crunch, very pretty sprinkled over the top of an omelette too.
Folding: You can leave your omelette open, likely the most historic version. Most people currently fold theirs once, to make a half-circle, as I do. If your tribe has strong French influence, however, you may prefer the double-fold version instead.