Similar to Hidden Grottos, my region Atlas has pokemon Nests instead! When you visit them, there might be an abandoned pokemon egg for you to take! The pokemon will be a surprise
I'm thinking in some cases maybe having a pokemon guarding the egg that has to be defeated first (which you could also catch). Or instead, there's some "mean" pokemon with bad intentions that you have to save it from.
I always thought eggs were a great idea but underused. Most eggs you'll ever have are ones you've bred yourself, and you know exactly what they are and you're doing it usually for battlling. This way you have a chance of getting special pokemon and the fun of not knowing what it's gonna hatch into!