I must begin my answer with a disclaimer of sorts. In my part of the world, I have had few opportunities to witness to believers in Islam. And, the few times I have spoken to a Muslem, I have not had visible results. Others can help more in the practical side of this needed area. However, during the last few years I have studied Islam more and more. I am no expert, but I am beginning to understand things that I did not know before. I also know that my feeble thoughts might give someone else an idea that they can develop further. So, in the hopes of adding to the "conversation," I will make some comments.
One thing that has jumped out at me as I have studied the teachings of Islam and the Quran is how carefully these teachings harden the believers of Islam against the gospel of Jesus Christ. In a sense, they are given a strong inoculation against believing the Bible and salvation through the death, burial, and resurrection of Christ. Islam teaches that the Bible has been hopelessly perverted, that Jesus was never crucified or resurrected, and that the teaching of the trinity is polytheism (a belief in more than one God). They are also taught that their sins are forgiven by the declaration of God when they submit to His ways (as taught in the Quran). When you take the standard salvation message to those who have accepted such teachings considering their rejection of the Bible, of the gospel of Jesus Christ, and of Christianity as a false religion, it makes it very difficult to begin any conversation about spiritual things. However, there are ways that true believers can certainly be a witness.
First, emphasize and demonstrate your personal relationship with the personal God. The Allah of Islam is very cold and harsh. It is true that the Quran emphasizes His mercy, but even this is a cold straight-forward act of a distant god. On the other hand, the God of the Bible has a very personal and intimate relationship with His people. He is our guide and counselor. He is our friend and helper. He speaks to us through His word and we speak to Him personally through prayer. The prayer life of a sincere Christian is very warm and heartfelt. He or she does not just say prayers, but talks to God about every kind of need and problem. Ask the Muslim about their relationship with God. Tell them about your relationship and demonstrate it in your life. In a sense, their god is the unknown god while the true God is known by His children (see Acts 17:22-28).
Second, live and demonstrate the love of Christ that true believers have for one another. Jesus taught His disciples that this should be a drawing point of their faith. John 13:34-35 states, "A new commandment I give unto you, That ye love one another; as I have loved you, that ye also love one another. By this shall all men know that ye are my disciples, if ye have love one to another." Explain that many who claim to be Christians are not, but that true believers have a love for one another that crosses all other boundaries. Note: the problem with this is that many so-called Christians do not demonstrate this love at all. But it should be something that commands the attention of others.
Third, read the Quran yourself and ask questions about it. As you read it, you will find many things to ask questions about. I do not recommend this unless you are strong in your own faith. However, most Muslims, though they give great lip-service to the absolute perfection of the Quran, know very little about it. Careful reading of it brings up many questions that can be starting points for spiritual conversation.
Fourth, use the Bible constantly. I know that they reject its authority. However, the Bible is so powerful that it defends itself. Show how the Bible has answers for life's problems. If your are speaking of finances, give a Bible verse on that. If you are dealing with child rearing, give Bible teaching on that. If you do not have the knowledge of the Bible to do this automatically, then look up verses on the topics you have discussed and bring one or two up the next time you meet. Allow the Bible to work in their lives and hearts. It is powerful (Hebrews 4:12) and can do a great work.
Fifth, introduce Jesus Christ. The Muslims think they believe in Jesus but simply reject His deity. However, they do not know Him. Ask them what they know about Jesus. At first, instead of arguing about the crucifixion or resurrection or similar topics, talk about Jesus as a man and as a teacher. Talk about some of His parables. Jesus came to present the Father to men. Even though Muslims reject the biblical teaching concerning Jesus, you can still talk about Jesus with them. Much of witnessing is found in introducing people to Jesus.
Sixth, use the Bible pattern for conviction found in John 16:7-11, where the Holy Spirit convicts men of three things: sin, righteousness, and judgment. Sin refers to the sin they commit. Is it really forgiven? How do they know? What do they think is sin? Do they know what the Bible calls sin? Righteousness refers to the righteous life they should be living. What would a righteous life be? How do they know? Show them how their ideas compare with the Bible teaching about righteousness. Judgment refers to the judgment every man and woman deserves. Are they sure they will not have to face hell? Why? Why should God just forgive their sins? All of this will help to bring them to Holy Spirit conviction.
Seventh and last: pray, pray, pray. Understand that you are fighting a spiritual battle at each step of the way. Islamic law calls for the death penalty for any Muslim that converts to Christianity. Although this person may live where this is not an issue, it will help to reveal the tremendous pressure to stay a Muslim. Their eyes are blinded by the god of this world (2 Corinthians 4:3-4) and you are trying to instruct those who oppose themselves that they might recover themselves out of the snare of the devil (2 Timothy 2:25-26). The Lord must do His work in order to show them the truth of His precious salvation.