Have them pull out their smart phone and open up Google maps and touch the icon to zoom to their current location. Remind them that what they just did is only made possible by satellites in orbit using clocks specially tuned according to general relativity to maintain accurate synchronization with clocks on the surface (i.e., in the smart phone). Welcome them to believe in a flat Earth as long as they remember they are lucky that everyone involved in making GPS work believes it's an oblate spheroid.
Have them travel to a dark spot in the northern hemisphere and observe and partially catalog the night sky. They should make notes on the big and little dipper, Orion, Cassiopea, and other major constellations. Have them travel to a dark spot in the southern hemisphere and repeat the process, noting that none of the familar constellations are visible but all new ones like the Southern Cross are visible. Have them return to the north one more time to verify that it is their travel that made the sky look different. Have them attempt to explain this with a flat Earth.
Many refutations on the Flat Earth Wiki rely on the "vanishing point" of perspective. The assertion is that things don't disappear from sight because of a curved Earth, but because of them "sinking to the vanishing point". Commision a helicopter ride on a calm day. As the helicopter rises, point out that while perspective makes the helipad seem smaller with distance, it doesn't move laterally through the field of view.
Do they really think that a secret known to more than 500 people can be kept for more than 50 years? https://en.wikipedia.org/wiki/List_of_space_travelers_by_name
If the Earth is flat, the entire American space program was/is fake. If that's the case, why would the Russians, Chinese, Iranians, North Koreans, etc, continue to say nothing about the American space program being fake? Why would they all strive to create their own space programs and ballistic missile systems? Perhaps some nations, like Russia, might also have a vested interest in the conspiracy, but not Iran. Not North Korea. Not Venezuela.
In history, people who have made demonstrably true realizations and discoveries that are inconvenient or contrary to those in power have been persecuted. Why are no flat Earthers persecuted? Why does the Catholic Church not support flat Earthers? Why does everyone pretty much ignore them?
I think a flat Earther might have said it best:
...even cults seem to all be ball earth.
Go with your friend and re-create all of the experiments from the seminal flat Earth book, starting here: http://www.sacred-texts.com/earth/za/za06.htm and using this math for calculating what should be visible: https://en.wikipedia.org/wiki/Horizon
I'm pretty sure your friend will prove to themselves that the alleged results of these experiments from Earth Not a Globe are actually falsely reported.
Alternatively (as in experiment 7) the alleged expected result for a round Earth from the experiment is falsely reported. For example, the author wrote that if the Earth were round, the horizon should seem to curve downwards left-to-right. Anyone considering a globe for only a few minutes could see that makes no sense. Suppose the round Earth expectation were true. Would not the necessarily fake Moon landing footage have showed such a curvature? Especially since the wrong-headed thinking that creates the expectation of curvature would further suggest the curvature expected would be greater for a smaller round body like the Moon. Yet the curvature was not faked in moon photos for the simple reason that we should not see curvature of the horizon on a round body as suggested by experiment 7. See: http://science.nationalgeographic.com/science/photos/moon-exploration-gallery/#/tranquility-base_1255_600x450.jpg
Really, just reading this book should be enough to do it. Chapter III begins in especially hilarious fashion by supposedly using science to prove the Earth does not rotate while instead actually showing the author knows nothing of science: http://www.sacred-texts.com/earth/za/za21.htm