I am shocked. An inaccurate Jeep fuel gauge...
Year would help here. The light is turned on by the cluster electronics. So, although it is pretty rare for the gauge itself to go south, anything can happen. Knowing what the gauge is reading and, knowing just how much fuel the Heep is capable of holding, go fill the tank.
If you are reading just under 1/2 tank and you put in 16+ gallons, there you go. Given that the average Heep holds 20.2 gallons, 15+ would indicate it to be 3/4 empty. The light, on my 98 at any rate, appears at the indicated 1/8 full level.
And, here is what my 98 FSM has to say on the matter:
"The PCM uses an input from the fuel gauge sending unit and internal programming to decide what fuel level message is required. The PCM then sends the proper message to the instrument cluster on the CCD data bus. If the PCM message indicates that the fuel level is below one-eigth of a tank for more than 10 seconds, the instrument cluster circuitry turns on the low fuel warning lamp and generates a single chime tone. Once the lamp is turned on, an increase in the fuel message of at least one full needle width for more than twenty secondes is required before the cluster will turn the lamp off."
So, what does all that jazz mean? I would start by confiming the fuel level by filling the tank and noting the actual level versus the indicated. If the two match, then the cluster electronics are suspect. Or the PCM...
If the Heep is OBDII, then the PCM can be read directly to see what it thinks. My FSM says to use the DRG tool but any quality reader will work. Provided it can actually access the sensors. If you wee closer, I'd say drop on by and we would fire up the laptop...
At any rate, post up the resolution when you find it.
I would reset the PCM and see if the issue changes. Just disconnect the battery and wait at least 10 seconds for the system capacitors to drain down. This returns the PCM to the factory settings. The PCM communicates with the Instrument Cluster over the Chysler Collision Data Bus. The messages it sends are read by the cluster electronics and all of the gauges/warning lamps are controlled. It is known as distributed intelligence in the computer world.