Somewhere between their first birthday and their first kiss, many children fall in love—with their toys. And it is a love affair that can last a lifetime.

For some investors and dealers, there is money in those old loves, as they look to reconnect people with the actual games that were consigned to memory (and the trash) so long ago. “When people want to relive their childhood, they will pay almost anything,” says Joyce Grant, founder of TimeWarpToys.com, an online vintage toy store that she operates from Phoenicia, N.Y. “It...