What was the exact age of Theon Greyjoy when he was brought to live in Winterfell with the Starks? How old was he when he was taken hostage? The age of 10 comes to my mind, but I'm not sure that this is correct.
Since you have asked this question for the Show, In Showverse, Theon was 8 years old when he was taken by Lord Stark to Winterfell.
Theon: I'm a Greyjoy. We've been lords of the Iron Islands for 300 years. There's not a family in Westeros that can look down on us, not even the Lannisters.
Ros1: And what about the Starks?
Theon: I've been Lord Stark's ward since I was eight years old.
Ros: A ward - that's a nice word for it.
Season 1 Episode 5 - The Wolf and the Lion
HBO Viewers Guide also confirms it:
As punishment and to ensure there would be no other uprisings, Robert forced Balon to surrender his only surviving son, Theon (then 8), for Ned Stark to raise as his ward at Winterfell.
1. Name corrected from Rose to Ros, thanks to Skooba
According to the novels, he was ten years old when he was brought to Winterfell.
"I was a boy of ten when I was taken to Winterfell as a ward of Eddard Stark."
A Clash of Kings - Theon I