That's an interesting idea...... I suppose we already do that for our house guests. LOL
We label each egg with the hen's name & date made. Our neighbors (who buy our eggs) stop by the backyard and thank the hens by name. LOL "Oh thank you, Bubbles. Your egg had such a creamy yolk and was delicious." We even get requests for specific hen's eggs. The neighborhood kids like Trouble's and Cookie's cute little eggs, and the guy behind us enjoys Jewel's big meaty Orpington eggs. (Those and the blue eggs from Tyrion are my favs.) To add variety, sometimes I'd slip 2 quail eggs in place of a chicken egg. We seriously had too many and there's only so many we could eat. I never labeled the quail eggs, so the best I could do was pull one out for a kid to pet.
I saw one egg company (organic, free range, pastured eggs that costs $9/doz at Whole Foods) which did something similar. They inserted a little slip of paper with a pic of their "hen of the week" and a little description of her breed, likes, etc. It was a cute marketing gimmick.
I suppose the closest thing to your restaurant idea would be "White Fence Farm" in Romeoville. I've never actually been there, but I was told they have a petting zoo at that location. I think it's a little closer to you, but perhaps we should meet up there someday.