He didn't really ask for cheap. He asked for the best chances of survival. Unless the lockbox is airtight and humidity/temperature controlled, your cheap DVD media could degrade over time.
The jury is out on how long it would take for optical media to degrade to the point of data loss, as many tests seem to yield varying results depending on the quality of the media used, but depending on the kind, (writeable vs rewriteable, etc), the general consensus seems to be that the survivability of most optical media in average room temperature and humidity is several decades, (citations obviously would be handy, but I'm not up for finding them). Archival grade optical media is your best bet if you head down this road, and it's not necessarily cheap.
Flash media may be even more volatile than optical media. Without power, minute leakages over time will lead to the loss of data, possibly within months or just a few years. (Again, citation needed.... I have the flu, so research this yourself to confirm... this is just a guideline for investigative consideration).
Tape media may still be a good bet, and probably better than magnetic HDDs. Tapes are small, store lots of data, and are pretty resilient.
I wonder how long they require/expect the data to last for? Years? Decades? Generations?