What is more important than this, and at the same time more terrifying? Being able to effectively talk about Jesus to relatives really requires a type of permission. You earn this permission by the life you lead. The tough part is that close relatives get to see you at all times–including your worst. A Christ-like example is your best witness and almost a pre-requisite to talking about Jesus. I say almost, because there are those exceptional moments that are set up by God. Moments where they ask you about Jesus, or the reality of impending death has forced the situation.
If your example is first then prayer is definitely second. Pray, pray, pray. Pray for God to soften and open their heart. Pray for boldness and the right words for yourself. Pray for others to be involved. Pray for love to overcome fear.
I don’t think every moment is a good moment to speak to relatives about Jesus, but we can sit back waiting for the right moment all our lives. The last thing you want is to be attending somebody’s funeral knowing that you never took the initiative to speak to them about this most important topic. In your prayers for your loved one, listen for God to give you an understanding of when to speak. Sometimes the moment is counter-intuitive, and you need to know what only God can know.
How and when to speak can vary wildly depending on the person and situation. Let love and the Spirit be your guide. In many situations, asking them about what they believe about death, the meaning of life, God, topics closely related to God, etc. is a good way to start. If you meet verbal resistance, let them know that the reason you are speaking about this topic is that you love them and you profoundly believe that Jesus is the way to eternal life.
Such a profession will either be met with silence or a volley of skeptical questions. Calmly answer to the best of your ability. Do not get defensive. Your attitude will say more than your words. Make sure that they clearly understand what the role of Jesus is in our salvation. Use scriptures like John 3:16, Romans 5:8, 1 Peter 3:18, and gauge how they respond to the Bible. If you see rising hostility or skepticism, ask why. If you perceive that they have reached a limit, thank them for listening and let the Holy Spirit work until a later time when you are called back into the conversation.
Some people are super uncomfortable with face-to-face conversations about God, especially with relatives. In such a case, ask them if they would read something or watch a movie. Gather good resources to suggest.
As painful as this is, remember that you cannot force faith on anyone. Don’t even think of disowning or writing somebody out of a will. Don’t force church attendance with a threat of some sort. That’s not love. Provide a good example, faithful prayer, caring conversation, well-chosen resources and don’t give up–even if you are requested to give up. In such a case, pray for somebody else to take up the mission for awhile and wait on the Spirit. Some of the most hardcore cases have been won over in the end.