As the system has been in for some time also check all the wiring and cable connectors. Chances are that there is a damage somewhere for example the Cable outer sleeve with the copper core exposed, Badly corroded connectors and as Spruce comments damage to the PCB.
A Fuse is designed to blow in approx 100th of a second in response to a sudden spike of Current and heat associated with a short. A corroded connector/Fuse or damaged cable would heat slowly as it has become a resistor to the current. The hotter it gets the more current is used to heat the cable.
The fact the fuse was slowly getting hotter is sign that there is damage somewhere in the wiring.. I would replace all the wiring as well as the Fuse/Fuse holder.
Look to use water resistant bullet or similar connectors avoid Choc block style connectors as these have exposed terminals that will get wet. Also avoid electrical tape on connections as this can pull apart over time. If possible protect cables in Conduit and keep cables runs as short as possible
