to grow weary: to be worn down by long effort until strength and resolve start to give out
This is the older, more vivid of the word's two lives. It pictures a person who has spent everything on a long, hard task and is now close to buckling. In the New Testament it is used for weariness of the inner person, not the body: the danger of losing heart and quitting.
The imagery behind it is athletic. The Letter to the Hebrews sets it inside a picture of a race, warning readers not to collapse like a runner who folds before the finish. The point is not ordinary tiredness but the deeper draining of will that tempts a person to give up on faith itself. In the message to the church at Ephesus in Revelation, the same family of meaning appears in praise: they have carried a heavy load for the sake of Jesus and have not worn out.
So the weariness here is a spiritual and moral condition, the kind that endurance is meant to outlast.