Hey Kool-Aid!

Kool-Aid Tie Dye Easter Eggs

https://www.myfoodandfamily.com/recipe/143963/kool-aid-tie-dye-easter-eggs

Basically, two packets of un-sweetened Kool-Aid (per color) and water. Instructions on the link above.  Reports are the scent fades (rats). Reports are also that certain colors work better than others, and that the eggs might taste like that flavor (sort of gross, depending on the flavor).

I haven't tried this one, but I have tie-dyed entire t-shirts and that works, so why not eggs?

Watercolor Eggs - Mighty pretty, but don't eat 'em

OH MY! Creative features a post on using regular watercolor paints (her dad's) to decorate eggs. She paints them, uses plastic wrap with watercolor to do a texture, and also bleeds them.  Beautiful stuff!

However, these are "just for looking at."  You might even want to seal them.

Here's the link:  Watercolor Easter Eggs - A New Way to Decorate Eggs