In Game Description
A large petrified egg.
Surely bears no life.
Eggs are vessels that harbor life itself,
and symbolize the deepest secrets of existence.
But what does a petrified egg harbor?
If it's not given to Magerold it will remain in your inventory when going to NG+, making the Dragon Convenant available at an earlier point.
If it's in your inventory, you'll be unable to use Magerold's services for that game cycle unless you give it to him (he'll keep asking if you give it to him, refusing him ends the dialogue).