Martyr is from μαρτυρ, the Greek word for “witness”.
In the early Christian Church, telling other people about Jesus (and if they thought you were a Christian) was behavior that could result in punishment including death.
If one wants to be a martyr for Christ, I think one needs to have an attitude of wanting to tell others about the gospel (good news) of Jesus Christ. I think this is best done in a state of grace and by being empowered by the Holy Spirit, with love.
At Mass tonight (on another topic), our priest said it is important to have an attitude of love. Sometimes love must be tough love. And if love must be tough, it must be so in the best possible way.
I think the words “Go in peace to love and serve the Lord” said at the end of Mass are among the most important words of the Mass. We must have an attitude of being changed, being filled with Christ and being willing to do something by offering ourselves as members of Christ’s body here on earth.
I heard a former NFL football player speak about a week ago (at a small meeting of perhaps 150 people). Although he had three super bowl rings, it turned out that this former football player was really a fine man with admirable character. He emphasized that we must endeavor to have three "C"s. Character, Conduct and Conversation. As Christians, we really should strive to show ourselves bearing the Fruit of the Holy Spirit. And have hearts that are aware of (and not stifle up) the needs of others.
Perhaps the greatest need that others have is for Christ. And yet we must also show concern for other dimensions of humanity in order to serve Christ in others (I think this is what Mother Theresa did).
There are those who were witnesses for Christ in the midst of extreme dangers and yet they were not killed during their most challenging encounters. Such as Bishop Pacelli (who later became Pope Pius XII), and Saint Anthony. Just because they encountered extreme evil and were not killed (spared by God because their work was not yet finished) they aren’t considered martyrs. And Mary, the mother of Jesus, is considered to be the Queen of Martyrs. Yet she was not directly killed by others because of her witness. Aren’t these (Bishop Pacelli and St. Anthony) also examples of martyrs? I think so. They were among the best witnesses for Christ the world has ever seen.