Can a woman’s period save her life years later? A company called Cryo-Cell International says that it can — that menstrual fluid contains stem cells that might one day be used for medical treatments. The company has not published research verifying the claim. But using the slogan “Your monthly miracle,” it has begun offering, for a fee, to collect and store cells from the fluid for a woman’s future use. Cryo-Cell, in Oldsmar, Fla., is one of several companies trying to make a business out of banking stem cells. Although businesses that store umbilical cord blood have operated for years,...