What are some of the most compelling issues concerning the Christian faith that the skeptic will be forced to consider as a consequence of reading Mere Christianity, by C.S. Lewis?

It's first important to note that Lewis, basing the book on radio talks aimed at a mass audience, doesn't stray from  standard Christian theology. "Mere" Christianity is orthodox Christianity. As such it challenges skeptics in essential ways. At its core, what might be most compelling to the skeptic is not Lewis's theology but his method: he is using logical, rational argumentation to show that, in his opinion, Christianity makes sense and is compatible with a rationally ordered universe. He is not saying one merely has to accept Christian doctrine because "the Bible says so." Even when he says he accepts certain Christian doctrine on "authority," he offers a chain of logical reasons for doing so. Relentlessly throughout the book, he offers reasoning the skeptic can think about and argue with. Among other claims, Lewis asserts the following:
That moral law exists as fully as physical law as part of the fabric of universal reality and that it proves that God exists. Lewis says that we humans have a universal sense of right and wrong and a yearning to see justice enacted that can only have been implanted by a deity. Our "natural" sense of moral right and wrong thus proves God's existence.
That Jesus Christ really was the son of God and not merely a charlatan or madman "lunatic." Once again, Lewis lays out a logical, three-part argument for this and offers rational reasons for choosing the first option. 
That Christians are imperfect but that fact doesn't negate the faith: "a Christian is not a man[sic] who never does wrong ..."
That God did come to earth (become incarnate) as a human being to atone for human sin and that this worked. Lewis likens atonement to nutrition. We don't know exactly how either one works and there are rival theories but we all agree "if you are tired and hungry, a meal will do you good."
That humans cannot fully grasp the logic or the magnitude of the divine gift of atonement, but can accept it. This is an arguably post-modern argument that our logical constructs are kluges with limitations that ultimately collapse under the weight of their own contradictions--but Lewis argues, the truth is still out there and we can accept it gratefully, if not with full comprehension.
That human traits such as pride and love are spiritually undergirded: pride is the great sin ("the utmost vice") dividing humans and love ("a state not of feelings but of the will"; the determination to act charitably) the great secret holding the universe together. A skeptic might find it compelling to think about how simple, concrete behaviors could have an outsize impact on the cosmos.
In the end, Lewis's Christian claims about the reality of God, sin, and atonement might not persuade the skeptic, but his logical argumentation provides a framework that offers food for thought.