The idea was to raise Theon as a member of the Stark family, so that he would come to understand and respect the ways of Winterfell. That included allowing him to ride around the woods and handle weapons. This was common practice with ancient and medieval hostage taking. From http://en.wikipedia.org/wiki/Hostage :
The practice of taking hostages is very ancient, and has been used
constantly in negotiations with conquered nations, and in cases such
as surrenders, armistices and the like, where the two belligerents
depended for its proper carrying out on each other's good faith. The
Romans were accustomed to take the sons of tributary princes and
educate them at Rome, thus holding a security for the continued
loyalty of the conquered nation and also instilling a possible future
ruler with ideas of Roman civilization.
Theon could have tried to kill Eddard or one of the Stark children to avenge his brothers. But if he did so, he would probably be hunted down by the surviving Starks or their bannermen. It's a very long way from Winterfell to the Iron Islands; Theon's chances of making the journey as a fugitive who was wanted for murdering a Stark would be rather slim.
Not only that, he would have dishonoured his father and likely provoked a war between the Iron Islands and the rest of the Seven Kingdoms -- a war which, prior to the death of Robert Baratheon, the Iron Islands would almost certainly lose. So he would have been less than welcome if he reached the Iron Islands. Theon knew all this, and in any event he seems to have liked and respected Eddard and his children, having been taken to Winterfell at a very young age.
Finally, there is no indication in the books that Eddard fully trusted Theon. Eddard was never alone with Theon, and he didn't send Theon far away from Winterfell, much less order him to visit the Iron Islands. As Thundergr points out, that was Robb's mistake.