There are many things we can do to help a non-Christian friend who is considering suicide. One of the most important things is intercede (pray for them) with God on their behalf. This is no small thing, add-on, or something to do when we have tried everything else. It is the first thing we should do in most situations (I Thessalonians 5:17). In everyday life, what do we do when faced with a situation that is beyond our ability to handle? We call someone stronger, wiser, or better equipped to handle the crisis. That is why we pray. We pray because God is stronger, wiser, and fully equipped to handle any situation. By taking the matter to Him in prayer, we are exercising our faith in Him and our trust that He is able to do above and beyond anything we can do, think, or imagine (Ephesians 3:20). God is our strength, an ever present help in times of trouble. There are no hopeless cases with God because nothing is impossible for God (Luke 1:37). There are many examples throughout history where the persevering prayers of faithful believers have led to the conversion of people who might have been considered hopeless or helpless (e.g. St. Augustine, who was the beneficiary of many years of anguished prayers made by his mother until he was finally converted). The prayers of a righteous man or woman can be extremely effective (James 5:16).
Second, share the good news of the gospel of Christ with them. For those who are lost and do not know Christ, suicide might seem like a rational way to escape the misery of a godless and hopeless life (Ephesians 2:12). If God does not exist, it is easy to imagine how one might fall for a philosophy such as nihilism. To counter this, you must present Christ to them in terms of HOPE. Christ is the way, the truth, and the life (John 14:6). He came that we might have abundant and eternal life (John 10:10). He gives direction and purpose to those who are frustrated with an aimless existence. He gives hope to the despondent soul who is withering in the slough of despair. He provides relief to the soul burdened with the pain and grief of guilt. In short, He is the answer to the questions, "Why am I here?" and "Does my life have meaning?" God has answered those questions by providing His Son, who is the purpose and meaning of our lives. Our purpose is to live to glorify the One who died for us. Christ delivers us from the penalty of sin (eternal death) and the power of sin (living here and now in a state dominated by our sinful natures). The wages of sin is death, and for those living under the burden of sin, suicide seems like a way out (Romans 6:23; Matthew 27:5). However, to die without Christ is not to be free from problems, but to increase them (Mark 9:48). There is no returning from suicide. It is appointed once for a man or woman to die and then face the judgment (Hebrews 9:27). Therefore, it is of eternal importance to settle the state of one's soul here and now. There is no reincarnation. There are no second chances, paroles or purgatories. This may all sound very ominous and depressing, but I think it's important to impress upon people who are considering suicide that it will not improve their lot. It will not decrease their pain. Christ, not death, is the key to joy. So, focus on the hope that is offered to those who believe on and receive Jesus Christ as Lord and Savior. Share with your friend what Christ had done in your life. Share how He has given you forgiveness, peace, joy, and life!