Instead of peeling the eggs before immersing them in the beet liquid, I cracked the shells so the liquid would seep in through the cracks and color the eggs irregularly.
Instead of cooking beets so I could use their liquid, I used beet powder that I found in the bulk spice section at Whole Foods.
I picked it up a while back, knowing that I wanted to use it to color something, but I hadn't quite figured out what I wanted to color. Eggs seemed perfect. Particularly since I wanted to color the eggs themselves and not just the shells.
1 tablespoon salt
2 tablespoons beet powder (or more, if desired, for darker color)
2 tablespoons white vinegar
Place the eggs in a saucepan and cover with cold water to cover the eggs by at least an inch or two. Add the salt. Bring the water to a boil, then turn off the heat and cover the pot. Let it sit for 10 minutes.Drain off the water so it's just enough to cover the eggs.
One by one, remove the eggs with tongs and crack them gently in several places and return them to the pan. Add the beet powder and the vinegar, stir to dissolve the beet powder, and let the eggs sit in the warm liquid for another 5 minutes. Transfer the eggs to a storage container, add liquid to cover, and refrigerate.
The longer the eggs sit in the liquid, the more the color will seep in through the cracks to color the whites. I left mine overnight.
If you like, you can peel off little bits of shell where you want brighter splotches of color.
Peel the eggs, and serve as desired.