Speaking of his apostolic ministry, Paul says it is by God's mercy that we are engaged in this ministry (2 Cor 4.1) of evangelism to the nations. This is a very interesting thought; I think it goes against the tendency of our thought at times.
Evangelism can seem more like a duty and a burden than a gift, something given by God's mercy towards us. After all, we have to tell people to change their lives, that in spite of how things seem to them, they are actually headed in a completely wrong direction. This can be uncomfortable in itself, but it is doubly so when the culture at large considers evangelical Christians to be these backwoods, bigoted know-nothings who deny science, who want to impose their sexual values on the rest of us, etc. All of this makes evangelism a burden, and sometimes we can be loath to do it.
Paul insists, however, that it is by God's mercy that we are able to bring the good news of salvation to others. And this makes sense, since it is by God's mercy that there is a good news of salvation in the first place! God saved humanity in Jesus Christ out of his mercy and goodness, not because we deserved it or anything of the sort. In the gospel of Christianity the focus is always on God's mercy and goodness towards the undeserving.
More than that, evangelism is a calling. God told Abraham that his offspring was to be a blessing to all the nations (Gen 22.18). We know that Christ is that offspring (Gal 3.16) and the blessing he gives is salvation, justification, life with God. But Paul insists likewise that if you belong to Christ, then you are Abraham's offspring (3.29), and so we must be a blessing too! The blessing we give is not any salvation we've accomplished ourselves, but the message of Christ who has given himself for the life of the world (2 Cor 5.14; cf. John 6.51).
I think part of the difficulty of evangelism may come from employing an unnecessarily contentious and polemical style. Karl Barth says in Church Dogmatics IV/1 that the fundamental Christian message is "God with us" (p. 6). That is a perfectly fine evangelistic message to give people, one that adequately summarizes Christian thinking without being unnecessarily complicated or esoteric. We as Christians believe that we have fellowship with God through Jesus Christ, that he has given us this gift so that we can live forever with him, and we invite anyone and everyone to come and participate with us. It may invite further questions but it doesn't unnecessarily raise them from the start. Moreover it is an irenic rather than polemical message, one that will not necessarily put people on the defensive. At the same time it emphasizes that Christian religious life is a matter of faith -- We as Christians believe -- and so does not make a claim to knowledge impossible to justify.
A friend recently told me the story of when he saw a young guy trying to evangelize some teenage girls at a Starbucks. He was using Greek terms and mentioning sophisticated principles of theology. Then another who was present, evidently convinced that this would ultimately be for the good, decided to interrupt and correct him on the use of some of these terms. Then the two of them debated between themselves while the girls left. This kind of thing is utterly inept and ineffective, like those campus preachers who shout at fornicators, masturbators, etc.
Much better is Christ's message: the kingdom of God has come near (Mark 1.15). God is with us in Jesus Christ, and he invites all to come and have fellowship with him.