Some possibly exotic ideas, in decreasing difficulty:
Warp drive: Specifically, explain the Alcubierre Drive (
http://en.wikipedia.org/wiki/Alcubierre_drive) in really basic terms, and point to an experiment that NASA is conducting on warp fields (
http://en.wikipedia.org/wiki/White-Juday_Warp_Field_interferometer). This is a somewhat controversial area - but not ridiculously so. =)
Simulation hypothesis: The idea that the Universe is a simulation, possibly made by our descendants. This doesn't seem like physics - until you read this paper:
http://arxiv.org/pdf/1210.1847v2.pdfQuantum computing: Talk about a) what it is, in simple terms of course, b) main difficulties with making a real quantum computer, c) discuss D-Wave Systems - a company that claims to have made a usable quantum computer. Discuss the controversy there.
Magnetic monopoles: These are things that shouldn't exist in standard electromagnetism (i.e. Maxwell's equations) - but if you're prepared to change definitions a bit, you can make a case that they've been discovered (
http://www.sciencedaily.com/releases/2013/05/130531103910.htm).
The Kardashev scale: A way of classifying civilizations in the galaxy, based on their available energy. There's plenty of easily digestible material on the internet for this - e.g. Michio Kaku talks about this often. It's also really nice because you get to talk about aliens in a physics presentation.
This is all just off the top of my head - let me know if you need more links/ideas/explanations. =)