Andrew, there's no 'normal' way to do this. Pretty much whatever
fits in with your thinking and scheme will work

Storage in either EEPROM or Flash will do, although that will
take some milliseconds, so you'll have to buffer incoming UART
characters in RAM. And if they're in RAM, not much point storing
them somewhere else as well if they're going to be discarded

Display drive can be kept going whilst incoming data is picked up
on interrupt from the UART. Storage in EEPROM/Flash on the fly
may be a problem because IIRC program execution is suspended
during a write, or a least for part of it


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