It's a metaphor that originated in the 19th century, when stage lighting used limelight, a bright light produced by heating a cylinder of lime. Actors who were "in the limelight" were those who were on stage, receiving the most attention.
Here's how the connection works:
* Limelight: A type of stage lighting used in the past.
* In the Limelight: To be the center of attention, similar to how actors in the spotlight were the focus of the audience's attention.
So, while "limelight" and "limestone" share the word "lime", they have completely different meanings and origins.