As others have already said, interstate transfers (other than an exception for bequests/inheritances) must go through an FFL to avoid violating federal law.
Here are your options:
1. Buy the gun and take it to FL yourself. Then you and she can go to an FFL in her area and that FFL can transfer the gun to her. The FFL will charge a fee.
2. Buy the gun and then ship it to an FFL in her area. You can ship to the FFL directly as long as you are sure that (1) she will pass the background check and (2) the FFL in question accepts shipments from non-FFL holders and is expecting your shipment. They are not required to accept such shipments and many do not. If it is a handgun, you can not ship it through the U.S. mail and the two shipping companies that handle handgun shipping require that you pay for overnight delivery. There will also be a fee from the FFL in her area to handle the transfer to her.
If she fails the background check, then you're in kind of a pickle. The FFL can not return the gun directly to you, it will have to be shipped to an FFL in your area and you will incur additional shipping fees and a transfer fee from the FFL in your area to claim the firearm.
3. Buy the gun, take it to an FFL in your area and have that FFL ship it to an FFL in her area. The FFL can ship via less expensive means, but will likely charge you a fee for handling in addition to the shipping fee. This will still likely be less expensive than you shipping it yourself if you follow the shipping companies' policies. There will be a fee from the FFL in her area to handle the transfer to her.
In this case, if she fails the background check, the FFL in her area can ship the gun back to the FFL that you had ship it. You'll still be liable for shipping and any additional handling fees.
4. Send her the money and let her buy her own gun in her area.
Do NOT go to the store with her and have her pick a gun, let her fill out the paperwork and then step up and pay the money. While this is legal, strictly speaking, due to the circumstances of the situation, it will likely arouse suspicion on the part of the dealer who could decline to complete the transaction at that point.