How to do it:

1) Be on Disc 4 and enter Madain Sari without Garnet or Amarant in the party. Lani, who usually runs away if said two are in the party, will stay and you can talk to her. On the second time, she'll mention that she saw some strange writing on the Eidolon Wall that might interest Garnet.

2) So, how does one find out about the walking around part? In the room by Lani, one can find an inscription that talks about the pattern which, if you can't decipher on first glance, goes likes so: enter the Eidolon Wall, head left until you come to the entrance and heard a 'ping'; then, walk back the opposite way until you come to the entrance and hear another 'ping'. Repeat until you get a message that your HP/MP has been refilled (9th time). On the tenth time, new writing will be available.

3) The new writing will tell about Garnet's past, her name, and the wishes of her surviving family member (I think. Haven't played in awhile).