Asha was a Cardassian war orphan who was abandoned on Bajor when the Cardassian military left following a lengthy occupation. She was looked after at the the Tozhat Resettlement Center by volunteers like Deela.
In 2370, Doctor Julian Bashir and Elim Garak visited the center in search of information on the Cardassian orphan Rugal. Asha politely asked if Garak had come to take her and the other children home. (DS9: "Cardassians")
Asha's name was not mentioned in dialog. Her name comes from the script, which also describes the moment she sees Bashir and Garak; "A fourteen-year-old Cardassian (ASHA) wearing clean but distressed Bajoran clothes quietly ENTERS, staying along the back wall. There's a cautious look in her eyes: wary but wanting, hesitant but hopeful.." This information would also place her birthdate sometime in 2356. 
The Pocket DS9 novel A Stitch in Time (by Andrew Robinson) reveals that, in 2376, Garak has a nightmare in which Asha and the other children appear asking the same question, if he was to take them home. Garak, waking from his nightmare, breaks down shouting that there is no home anymore, Cardassia having been devastated by the Dominion.