"The character possesses a remarkably good horse sense; when following a plan of this character's making, one die roll made by anyone, including the DM, can be adjusted 2 points in the character's favor, one time per day."
Essentially, there's a chance (I'm not sure what the chance is, yet, I'm still adding things) of picking up this 'ability' at the start of the character's creation. It does require a little more description than what the above includes, in order to clarify what's being said and to narrow down the effect somewhat.
"Horse sense" is an idiom that developed to convey an "unsophisticated, country type of sense." The link makes the connection that horses are often sort of dim, being associated with words meaning nonsense of stupidity, such as horse feathers or horseshit (a dumber form or shit than most others). Having been around horses just a little bit, I feel the term derived from a sort of 'sense' that horses have about bad paths, bad footing or just places they don't want to go - places where the rider overrides the horse's will and finds out the horse was dead right about not going into that hidden cactus patch.
In any case, a horse is always dumber than the dumbest human, so any character of any intelligence fits into the category.
"A plan of this character's making" would be any plan that was originally proposed by the player, which is now being followed. It can only include the player's ideas specifically - not the ideas of others who advanced the original plan by adding features and corrections.
Thus, the player with the ability says, "I think we should dive into the pool and see what happens." If the character with horse sense or any other character actually dives into the pool, then the +2 bonus can be applied in that situation. If, on the other hand, a character suggests easing into the pool, paddling around for a bit and then going underwater, that would not be the plan proposed and the bonus could not be applied.
It must be remembered that the bonus is not applied to the character making the plan, but to the plan itself - though the character with horse sense does get to decide where the bonus is applied (saving throw, to hit die, damage done, amount of treasure found, etc.).
If the plan is carried out and the character does not think of using the bonus (or can't use it, as no dice are thrown) until after the situation is resolved, then the character must invent a new plan to regain the unused bonus for a new situation.
"Favour" expresses a positive result that encourages survival and success. The die roll cannot be used to bring woe or misfortune to another character (player or non-player).
Would I, as DM, accept the modifier at the player's behest? *smile* Absolutely!