1979 Ford Mustang Fuel Gauge Problems?
The most common cause of eratic reading/ nominal reading of the fuel guage is a bad "float" on the tank sending unit.
With age the float - like a mini brass hollow barrel develops tiny perforations which allow the float to get filled with gasoline and no longer float as it should. (see EBay - item 300180705079)
A float itself is inexpensive, about $10-12, but the work involved is considerably more. You likely don't need a new gas guage or a complete sending unit, this item is probably about $120, and most Mustang specialty shops can secure one for you ,if you really need it. I like CJ Pony parts, Vermont , USA.
One word of advice, unless you want your mother to lose a favorite child, DO NOT undertake this repair yourself. Gasoline fumes are very dangerous!