When I was a kid, we used to use toothpicks to draw geometric patterns on our hard-boiled eggs with melted paraffin. This acts as a mask when you dye them. You can even make multiple color patterns by starting with some wax for the original white, brown, or green egg color. Then use a light color dye like yellow (or just a quick dip in a dye). Then add more wax patterns, followed by a different dye. I have done four different dyes, resulting in five different color patterns, but the final un-waxed egg color ends up very dark.

Here's a link to a crafts web site with some examples:

Lithuanian Easter eggs

Ukrainian version

Paraffin is much cheaper than beeswax, and works fine. Also, it is translucent so you don't really have to remove it at the end.