Fill a large pot or bowl with buttermilk, salt, garlic powder, cayenne pepper, cumin, and lime juice. Whisk together.
Place Cornish hens in the milk mixture, using a spoon to coat the tops. Let Cornish hens brine for at least 3 hours and up to 8 hours, flipping the hens over half way through.
Preheat oven to 425 F. Remove Cornish hens from buttermilk and pat excess off with a paper towel. Discard buttermilk mixture.
Place Cornish hens in a shallow baking dish. Place lemon wedges inside hens, 2 wedges in each hen. Use cooking twine to tie up the legs. If cooking with potatoes, arrange around Cornish hens without covering the hens too much. Brush melted butter on hens and potatoes. Finish up by seasoning hens with salt and pepper to taste.
Bake Cornish hens in the oven for 50-60 minutes, until top is golden brown and potatoes are soft.