What purpose does the appendix serve?
Mainly to make evolutions look foolish, by holding it up as another PROOF of humans having come from lower life forms.
Thus, although scientists have long discounted the human appendix as a vestigial organ, there is a growing body of evidence indicating that the appendix does in fact have a significant function as a part of the body’s immune system. The appendix may be particularly important early in life because it achieves its greatest development shortly after birth and then regresses with age, eventually coming to resemble such other regions of GALT as the Peyer’s patches in the small intestine. The immune response mediated by the appendix may also relate to such inflammatory conditions as ulcerative colitis. In adults, the appendix is best known for its tendency to become inflamed, necessitating surgical removal.
Sadly, I've found this line of argument to not be effective, regardless of how logical it is. He can always claim that your idea of engineering "perfection" is entirely in the eye of the beholder and that any "design flaws" you point out are in fact eneffably purposeful and necessary parts of our makeup. The infection-prone appendix is obviously some sort of "test" that we must overcome in order to make us better people, along with a litany of other bad things that often happen to good people. It's all part of the big "plan".
The inevitable conclusion is that God did it because that's the way he wanted to do it, and it is not our place to question why. This is a supremely simple argument, but one that is entirely unsatisfying from a scientific point of view. If he did design us, he did so with the capacity, and indeed the drive, to seek out connections and recognize patterns in the world we live in. Evolution is just one of the incredibly beautiful laws we have discovered that seem to drive and organize the world around us.
Whether there was a supernatural impetus behind the existence of these laws is a matter of faith, but the patterns are there regardless of how you conceive they were originally wrought.