Whenever there's a PC involved, it makes sense to get the PC to do as much of the hard work as possible, and make the embedded processor's life as easy as possible.

If I understand correctly, the 8051 only ever reads the EEPROM - all writing is done by the PC.
Therefore, it would make sense to have the PC write the data to the EEPROM in a format that's easiest for the 8051 to read.
A file system as described would be very flexible, but sounds like massive overkill to me.

I would have the PC write the EEPROM using records something like this:

struct
{
   unsigned char design_name[40]; // Text string that can be displayed on the LCD
   unsigned int  number_of_rows;  // The number of rows in the data
   unsigned char data[];          // The variable-length data
}

(this wouldn't work as an actual 'C' structure, because they can't have variable length, but you get the idea).

When the EEPROM needs updating, the PC would read the whole thing, do the update, then write the data back as just a contiguous set of records - so there's no need for the 8051 to go following the chains of blocks making up a file