The way I see it, the "best" location for storing information depends on many factors. Each method has its pros and cons, and only by looking at the big picture can a decision be made. There is no one-size-fits-all solution.
This whole discussion looks at things from the point of view of the Yun's '32U4 processor. The AR3391 has completely different considerations.
For fixed data (including strings) that never change, the best location is usually in PROGMEM, especially if the data is accessed frequently.
However, if there is a large amount of data that does not fit in PROGMEM, and it is only accessed occasionally, then storing it on SD card, or requesting it from the Linux side may have appeal, even though there is some additional overhead to access it. But I wouldn't consider it unless it doesn't fit in PROGMEM. Storing such data in EEPROM would not be an option since there is little of it (and we're talking large amounts of data here.)
For smaller amounts of configuration data that changes rarely (mostly, but not completely constant) and which must survive power cycles, then EEPROM is the best choice. This data would be read on power up, and only written when it actually changes. Such data could be stored on the Linux side, but the extra overhead of fetching it is probably not worth it, and there is a significant delay before the Linux side boots and is ready -- EEPROM is ready right away.
If there is a larger amount of rarely changing configuration data, which doesn't fit in EEPROM, then storing it on SD card makes sense. Just keep in mind that reading it at startup would have to wait until the the Linux side has booted. A mixed scenario may make sense if there is a smaller subset of data that must be accessed shortly after boot (put that in EEPROM) while the rest that doesn't fit can be accessed later (put that on SD, or request it over the bridge.)
There is no single answer to the best type of memory to use for any particular system. You really need to do the analysis and make the determination based on the specifics of each application.
RAM: fastest access, limited quantity, loses values on reset or power failure
PROGMEM: slower access, higher quantity, changed only when loading sketches, retains values on reset or power failure.
EEPROM: very slow access, very limited quantity, can be updated but limited number of write cycles before failure, retains values on reset or power failure.
SD Card: very slow access, very high capacity, can be updated easily, retains values on reset or power failure.
Bridge: slow access, medium capacity, can be updated very easily, loses values on reset or power failure