A shine is a term for showing off, in this context in the center of the circle. While today, many shines are expected to be big and showy (e.g. aerials) many dancers in the 1930s to 1950s would shine with a silly move (which is a nice option if you're nervous about the spotlight). This may be a good time to show off that new move or movement that you've been working on (or just dance 3 circles and a swingout). Note, the circle is meant to be a place of community. Those on the circumference should support and cheer on those in the center.
Many dancers are intimidated by shines. It should be noted that, while some dancers may improvise their shine, many of the best dancers had a go to move or two. Start to develop a paired move or two with your favorite partner(s) as well as a solo piece. Again, they don't have to big and showy. As the Cromulons famously said "Show me what yo got!"
For inspiration:
Words that rhyme with Shine: line, design, fine, sign, nine, mine, devine, whine, wine, spine, vine, dine, enshrine, …
Note: you'll need to indicate who you're calling out to shine