Importune prayer is prayer that is aroused and energized by the Holy Spirit.  According to E. M. Bounds,

It [importune prayer] is not an impulse of energy, not a mere earnestness of soul; it is an inwrought force, a faculty implanted and aroused by the Holy Spirit.  Virtually, it is the intercession of the Spirit of God, in us; it is, moreover, “the effectual, fervent prayer, which availeth much.”  The Divine Spirit informing every element within us, with the energy of His own striving, is the essence of the importunity which urges our praying at the mercy-seat, to continue until the fire falls and the blessing descends.

 I think this is right on.  When we pray with importunity it is really the Holy Spirit praying in us.  He moves us and motivates us in our prayers.  Here are four ways the Holy Spirit motivates us to pray with importunity: