During Disk 4 you can discover Dagger's real name and her mom's name as well. To do so put Dagger in your party and go to Madain Sari. Go to Eiko's back porch and you'll see Lani but she runs away.

Now leave Madain Sari and form a party with anybody EXCEPT Dagger and Amarant. Go back to the porch and this time Lani will speak with you. She mentions a secret at the Eidolon Wall. So go check it out.

The wall is now presented as a sort of side scrolling room and if you go all the way in one direction you'll be back at the beginning. Go all the way in one direction and you will hear a chime. Now go back in the other direction. You need to do this, going back and forth, a total of 9 times hearing a chime each time.

On the 9th time your party's HP & MP will be restored and all status ailments will be removed. Now proceed back across the wall reading all the messages. There will be information about eidolons as well as messages that could be for Eiko. When you're done return to Ifrit's picture and an exclamation point will appear. Examine it and you'll find a message for Dagger and her mother that states both their names.