As often as the concept of Riddle not knowing of or feeling love throughout the story appears, it could be "assumed" that this power is to know or feel love.
Dumbledore himself labels this power as love. While, potentially, it could be the Deathly Hallows, this is unlikely as Harry does not possess them at the time his parents are murdered. It is not only the ability to love others, but the ability to accept love from others which enables him to defeat Voldemort time and time again.
I've always thought that the power is the ability to accept things, such as death or death of others.
The Power he Knows Not was grief, and it wasn't just Harry but everyone connected to Harry. Sirius, Severus, and Lily, chiefly among them.