She is not a proud person.
She doesn't do the human equivalent of baring her teeth and growling when her rank is ignored, like Sandry does; she isn't as set in her ways as Tris can sometimes be; she doesn't gloat over her special skills, like Briar does about his ability to pick locks.
Nevertheless, the day she receives her medallion is one she remembers with, yes, pride.
Because Frostpine grinned at her, crafty old man that he sometimes is, and said, "Now you can start teaching someone else to make nails." (If that isn't an achievement after all those hours of labor, Daja doesn't know what is.)
And Sandry looked at her with shining admiration, Tris with shocked pleasure, and Briar with shared excitement, the way she knows she eyed them and their achievements in return.
The medallion hanging around her neck embodies such an amazing day; how could she feel any less than pride?