Depending on how long you plan to collect data, you might want to look at the FRAM chips (Ferromagnetic Random Access Memory) - basically ancient core memory in a chip. You'll need an I2C capable PICAXE to talk to the FRAM but it is probably the static memory chip with the greatest number of read/write cycles.
If you want accurate time, you probably also need one of the RTC modules - they also speak I2C. The easiest to read data would be something like 210927.1013.n for on and 210927.1427.f for off. The "." is optional but makes it easier for humans to read if you display it. If you plan to send it to a comuter, just skip the "." unless you're putting it in a spreadsheet in which case that should be "," for CSV format.
Remember that the PICAXE could read the FRAM and spit out almost any format. In this format, year, month, date, hour, minute, on/off are each one byte - two more bytes if you store the "." or "," but the PICAXE can put those in as it reads out data from the FRAM, if needed.
You will have this programmed by the end of the day, won't you? ;-)