How does one send a box of food to Ghana without costing an arm and a leg?

There's no cheap way unless you find an organization which routinely sends packages and will include yours. I doubt they'll give it to a specific person.

There are 100's or more charities that operate in Ghana, your friend should contact them.

Also if you are a member of any major denomination, you could arrange the local church to furnish food in return for payment to the church.

Also if you want to send actual food, make it light containers, noodles, packaged rice, foil containers of dried or sealed meats and tuna. ( no cans)

Also there are even Kroger grocrey stores in Ghana, so perhaps arrangements directly with the store for food for them could be arranged.

Charity Christian Missions Ghana

Charities Working in Africa