The power children have, by repeated nagging, of influencing their parents to buy advertised or fashionable items.
This phrase emerged in the USA in the late 1970s; for example, this from The Washington Post, February 1979:
"'They (the children) use all the pester power they can muster’ to talk their parents into purchases, a narrator warned."
Children have long been able to influence what their parents bought on their behalf, by making it easier for the parents to give in than to endure relentless badgering. The fact that this wasn't a significant enough phenomenon to be given a name until the late 20th century is indicative of the changing nature of child/parent relationships in western societies. Parents are now more likely to take a child's demands seriously, whereas in earlier times they were expected to be content with whatever they were given. Also, the increased pressure on parents to work longer hours has led to a society that in increasingly cash rich and time poor, which makes it a more attractive option to buy their way out of the time-consuming task of fending off demands.