Dating Yuffie is a Project, since you need to know that you want to date her from the start of the game.
In case you didn't know, Final Fantasy VII keeps a record of how you treat the females in your party, and that is how it decides who you date at the Gold Saucer near the end of Disc 1. You have to start out knowing that you are going to try and date her.
Tips for dating Yuffie:
Be as mean as possible to Aeris and Tifa. For example
When you pick a party after Midgar, use Barret and Red XIII.
Also you have to get Yuffie ASAP -- right outside Junon. Then you have to use her for the rest of the first disc, and keep her second in your party. Don't use Aeris or Tifa in your party, use Red XIII, Barret, or Cid when you get him. Also, it helps if you give Cloud the cover materia. and he covers Yuffie.
Finally in order to date Yuffie you have to do all of her sub-quest before you go to the Gold Saucer. After you get the Tiny Bronco go to Wutai to start the quest. For more details see the walkthrough, there is a section on Yuffie's Sub-Quest.
When you get the date, it's really funny. I saved the file separately so that I could watch it a couple of times. It's the funniest of the three I have seen.