I feel that this statement always needs to be qualified depending on intent and context.
Something is strictly radioactive when it spontaneously emits radiation. Lunar surface material is not this. Your radium clock dial is. The dirt near where I live is, because it contains uranium ore. These kinds of substances are generally characterized by being composed of unstable atoms—uranium, for example. You can hold a moon rock safely in your hand for as long as you want. It does not emit radiation. It does not significantly contain unstable atoms. It is not radioactive in the strict sense.
As I've belabored, many substances including moon dust, earth dust, and a lot of metals emit radiation in response to being bombarded by an outside radiation source. When we blast beryllium with high-energy electrons, the beryllium reacts by emitting x-rays. This is the mechanism that lets the blokes at the airport see inside your luggage. Lunar dust and rocks do this. The materials that do this can nevertheless be made from reliably stable atoms.
I insist on the qualification because there are contexts in which this behavior is noteworthy. If you're planning to be in an environment were secondary radiation is significant, you must prepare for and protect against it just as if it were primary radiation. In the end it doesn't matter whether the x-rays come from the sun or from the lunar surface as a secondary emission. An x-ray is an x-ray, and you need to consider it as such when estimating biological effect—energy, flux, etc.—of being there.
You will find sources that say things like, "The lunar surface is highly radioactive." These statements are generally aimed at lay readers who aren't interested in full detail. But when you're talking about the authenticity of visits to that environment, you really do have to pay attention to the detail. You need to ask, "What do you mean by 'surface?'" and "What exactly do you mean by 'radioactive?'" These important distinctions apply to rebuttals of hoax claims, although not necessarily to non-technical summaries intended for casual readers.