The seven-year-old, who shares the same name as the Queens Park Rangers manager, will compete for the Under-9s next season.
Technical development officer Matty Holmes told Bournemouth's official website: "Harry was playing for Moordown locally so we had an eye on him.
"A few scout reports came back saying he was progressing very, very well, so we bought him into our academy sessions around November last year.
"He's been playing with the top Under-8s we've got and he's shown really good progression. He's improved his technical work and his receiving and passing skills have come on no end."
Redknapp junior mirrors the path of Harry's son Jamie, who came through the ranks at Bournemouth during his playing career.