I don't think there are ever super cheap ways to get to CMB, but I would try either Emirates or Etihad from JFK.

You can also try buying two tickets and breaking up the journey in Europe. I think for CMB, breaking the trip up in MUC or FRA is usually cheaper than LHR.

CMB isn't well served by LCCs, but if your daughter can get a cheap ticket to Dubai, she could transfer to Sharjah and take Air Arabia to CMB. That might be a bit of an effort, though, and would depend on how comfortable she (and you) are with arranging transportation from DXB to SHJ.