Focus on people's potential, not on their past. Peter's name change to Cephas (which means rock) is based on Jesus' recognition of his potential. He became a rock even though he was unstable. Man looks at the outer appearance, but God looks at the heart.
If you love someone long enough, eventually you break through by the grace of God.
Be ready to help when they seem to be sinking. Watch for teachable moments. He's looking for people who are willing to try. Peter called Jesus the Christ while he was among other "gods". So must we.
Give recognition and responsibility.
Never give up on people. As soon as Peter gets encouraged, he takes a step back. He let his confidence run away with him.
Luke 12:31 (beginning of any ministry): Jesus prays for Peter when Satan wants to sift him like wheat.