Postal regulations allow a non-FFL to ship long guns to an FFL. Handguns, however, must to FFL to FFL if you are using the Postal system. UPS and Fedex allow handguns to be shipped by non-FFL, but they both require the gun be shipped via their fastest route, so it may be cost prohibitive.
As per a conversation once-upon-a-time with an ATF field office. There is no regulation requiring that any kind of gun originate with an FFL for shipment. Some FFL holders will not accept from an individual as per their own policy.
I would suggest your friend strike up a relationship with an honest shop owner to just make the transfer process easy. For all shipping, the Postal Service is the best value. Handguns can go anywhere in the US for less than $20.