One of the strongest reasons for accepting dua is to make dua in the face of helplessness. Proof of this is the hadith about three people, where they were forced to take refuge in a cave at night. Later a rock came from the hill and closed the mouth of the cave. Then they said to each other, "Find out the deeds which you did for the sake of Allah alone, then seek Allah with the help of those deeds, then hopefully He will relieve you from here." Then they pray to Allah by their good deeds. As a result of this, the rock moved and they came out.

Reference: Bukhari: 2215