For avocado salsa, in a medium bowl combine avocado, red onion, cilantro, and lime juice. Season to taste with salt and black pepper.
In a large bowl combine eggs, the water, green onion, the 1/4 teaspoon salt, the 1/4 teaspoon black pepper, and the cayenne pepper. Beat with a fork until combined but not frothy. Heat an 8-inch nonstick skillet with flared sides over medium-high heat until skillet is hot.
Add 1 tablespoon of the butter to the hot skillet. When butter melts, add 1/2 cup of the egg mixture; lower heat to medium. Immediately begin stirring the eggs gently but continuously with a wooden or plastic spatula until mixture resembles small pieces of cooked egg surrounded by liquid egg. Stop stirring. Cook for 30 to 60 seconds more or until egg mixture is set and shiny.
Spoon one-fourth of the crabmeat across center of eggs. With a spatula, lift and fold an edge of the omelet about a third of the way toward the center. Remove from heat. Fold the opposite edge toward the center; transfer to a warm plate. Repeat with the remaining butter, egg mixture, and crabmeat to make four omelets. Serve immediately with avocado salsa.