The argument which McDowell calls the "trilemma" is popular among amateur apologists for Christianity. It was first popularized by C.S. Lewis, and has become even more common since McDowell reworked it. It is logically weak, but it is rhetorically powerful--as its popularity and recurrence attest--and so worth considering in more detail than it might otherwise merit.
The name "trilemma" is somewhat misleading. Traditionally a dilemma is a situation in which one is faced with two or more alternatives, each of which is somehow bad or unpleasant. "Trilemma" and the trifurcate phrase "Lord, Liar, or Lunatic" (LLL) suggest a three-way decision, two of which (according to the argument) constitute a dilemma, thus favoring the third. Structurally it might more accurately be viewed as a binary decision in which one of the branches is asserted to lead to a dilemma, thus favoring the other branch.
The original form of the argument as made by Lewis was ostensibly directed only at refuting the claim, sometimes advanced, that Jesus was a great moral teacher, but not God. In a nutshell: "If Jesus' claims are not true, then he was either lying about them (which is morally reprehensible) or he was deluded into believing them, which would make him a raving madman (whom nobody would respect as a teacher); thus he couldn't have been a great moral teacher." Lewis's version was originally for a radio broadcast, and is probably more properly construed as a rhetorical argument rather than a formal logical one.