The FLASH-MEM option was introduced in Moteino R2 to allow SPI flash memory (SOIC JEDEC 3.9/5.3mm body width package) to be added to Moteinos primarily for wireless programming (but can be used as non-volatile storage for any purpose). There is an SPIFlash library just for using this memory chip. The SPI CS select for this is connected to D8 on all Moteinos (D23 on MoteinoMEGAs).
The chip shipped on all Moteinos is the Windbond 4mbit W25X40CLSNIG. Higher capacities are available upon request for custom batches.
The DualOptiboot bootloader allows you to reflash the program from a flash image stored on the external FLASH MEM chip (wirelessly programming it). See this blog post for how wireless programming is achieved using DualOptiboot and more details. This is the pinout of the FLASH MEM chip used on all Moteinos:
Why flash chips and not SD cards that come in much larger sizes at low cost? Because of the size and power constraints, an SD card slot would increase the cost. A separate SD-card Moteino shield would be more appropriate for an SD card add-on.