I'd like to tell you SPECIFICALLY what all I experienced re the Stone Circle, but being as I had completed it a bit ago, I'd be hard pressed to do so. What I can say for sure is what you will be digging up there will take place at the first full moon AFTER UR Townies have completed the project of its restoration AND your town is visited by a mysterious stranger who informs you that you will be digging up something there. Where U need to dig will be as clear as where U need to dig for any other item such as stolen back notes, stolen minerals etc. All of which are indicated by a raised circle of dirt. To know when UR town will get its next full moon, all U gotta know is when UR In Real Life full moon happens since Virtual Town is played in Real-Time. PLUS YOU WILL BE VISITED BY THAT MYSTERIOUS STRANGER! If that has happened already and you've lost the opportunity to dig up this artifact before your mysterious stranger's ship had sailed away then I'd take a shot in the dark by saying this:
If the Stone Circle no longer has any indication of needing to dig up something, then your opportunity has been lost, but probably only until your mysterious stranger returns to visit your town again. And all the mysterious stranger's visit repeatedly.