To unlock this, you have to choose specific dialog options to get a hidden score with Barret higher than Aeris, Tifa and Yuffie. Some of the dialog will boost Barret up and some will lower the other 3. I am following a guide written here:
http://finalfantasy.wikia.com/wiki/Date_mechanicsSomeone scrapped through the code of the game to work out the score values from each conversation. To write a proper guide here, all I would be doing is stealing others works.
I have read multiple reports of this trophy not unlocking for people. The general consensus seems to be, if you over use the glitch in Cosmos Canyon that allows you to repeat a conversation over and over to increase Barrets score, then you will end up on the date, but the trophy may not unlock. Only perform the glitch around 10-20 times. It is thought that if you go over a certain score then it glitches, so if you follow the guide completely then don't bother glitching. If you don't follow it at all then around 20 times. Credit for the glitch calculations goes to Lilley1593 on another site. Make a back up save before glitching, that way if you get Barret, but it doesn't unlock then you can use the glitch less on another try.
I went over the scores if you choose the perfect choices and the numbers end up as:
Aeris 35
Tifa 33
Yuffie 28
Barret 54