A second childhood was the experience of re-visiting one's favorite activities from their actual childhood, or enjoying things one never got a chance to do the first time around.
In 2369, after Jean-Luc Picard, Ro Laren, Guinan, and Keiko O'Brien reverted to their child bodies, Deanna Troi encouraged Picard to think of his new youthful status as a second childhood, something many longed to experience, albeit without having to go through the painful parts of growing up.
Later, in Ro Laren's quarters, the Bajoran, whose first childhood had been anything but happy, didn't look at the changes that way, feeling violated instead. Guinan took a more optimistic approach, figuring that as long as they were kids, they might as well have fun with it. (TNG: "Rascals")