Of American-English origin, the phrase pester power denotes the children’s ability to pressurise their parents into buying something, or doing something for them, by continuing to ask for it until their parents agree to do it.

This phrase originally referred to television advertising targeting children—as shown by the earliest occurrences of pester power that I have found:

1-: From Kids Advertising Hearings to Open, by Larry Kramer, published in The Washington Post (Washington, District of Columbia) of Wednesday 28^th February 1979:

On Monday, five weeks of hearings on the merits of television advertising aimed at children begin at the Federal Trade Commission. But Action for Children’s Television (ACT), the Boston consumer group which first called for strict regulations of children’s advertising nearly a decade ago, decided not to wait for the hearings to fire its first volley.
So at a press conference here yesterday, ACT released its new 22-minute film examining the state of commercial children’s television, “Kids for Sale.”
[…]
Portions of the movie deal with the mission of advertising for children: to get their parents to buy a certain type of toy or cereal.
“They (the children) use all the pester power they can muster” to talk their parents into purchases, a narrator warned.

2-: From Maybe TV understands kids too well, by Jeanne Garland, published in The Sun (Baltimore, Maryland) of Sunday 7^th December 1980:

The cereal aisle of grocery stores and any toy store are places wisely avoided by parents when in the company of youngsters, especially this time of year. These areas can easily become noisy battle grounds testing the willpower of the parents against the pester power of the kid. And the more television the child watches, the more the odds are stacked against the parent.

3-: From TV important as school in child’s socialization, by Sharon Ward, published in the Hattiesburg American (Hattiesburg, Mississippi) of Friday 25^th September 1981:

“Television has become a device which is as important as the school in the socialization of children,” said Dr. Hans A. Baer, cultural anthropologist at USM, during his presentation at the Conference on the Influence of Children’s Television on the Child, Family and Society being held at USM today.
[…]
The sponsors of television programs expect the program to promote its products, whether through sex or violence or not, Dr. Baer said. The primary concern of a sponsor is to promote sales rather than to inform the viewers through the program being sponsored, he added. The same is relevant with programs aimed at children. The commercials that grasp the attention of children between cartoons have that goal of instilling in the child’s mind that product which, through “pester-power,” has a way of converting itself into cash for the advertiser.

The phrase has come to be used in British, Irish and Australian English, too—for example:

– The following is from The Guardian (London and Manchester, England) of Saturday 19^th June 1993:

GLOSSARY FOR THE NINETIES . . .
PESTER POWER n. The latest ad-agency buzzword: it refers to that sweet habit children seem to have of persuading parents to spend big bucks on specific brand names. Little cherubs.
                                        DAVID ROWAN

– The following is from the Irish Independent (Dublin, County Dublin, Ireland) of Thursday 13^th May 1993:

Parents suffer in pester-power play

Advertisers are increasingly targeting children to make the ‘hard-sell’, Eamonn Holmes explores the controversial strategy
Pester-power is the marketing and advertising industries’ latest weapon—using children to goad their parents into buying products. And the campaigns are not aimed solely at selling toys, breakfast cereals, sweets and soft drinks.
Orchestrated by the marketing men, it goes far beyond children’s classic nagging for sweets or a chocolate bar at the supermarket checkout.
Marketing firms believe young people are increasingly influential across the whole range of the family’s shopping.
So they are being targeted with goods that have no obvious connection with childhood, such as pet food, TVs, cars, stereo equipment, furniture and household cleaners.
[…]
Pester-power is a delight to the marketing moguls—it’s thought to be worth millions of pounds a year in potential sales.

– The following is from Give us a break, by Jane Freeman, published in The Sydney Morning Herald (Sydney, New South Wales) of Thursday 29^th June 1995:

A national survey on Australian children’s influence on spending patterns, issued last month by AMR Quantum Harris, found kids influenced the choice of holiday destinations in 33 per cent of cases (compared with their 81 per cent influence over fast food choices and 65 per cent over breakfast cereal).
The report found that children’s influence over holiday destinations had increased in the past three years. In niche marketing circles, the influence of children has been dubbed “pester power”.