Assuming you've removed all external connectors to isolate the problem is on the mpu/driver board?

What you have is a resistive(almost-shorting) load pulling down the 5v, hence it becoming 2.4v.

Since it happens over a period of time instead of instantly, i'd start by changing the filter cap on the 5v supply, also including any filter caps (large or small) that are on the supply that feeds into the 7805 or lm723 voltage regulator.
Capacitors after a long time will become resistive to DC and start becoming a short (or resistive-short) instead of being somewhat open-circuit to DC voltages.
There's probably still enough chemical(electrolyte) inside the cap to initially startoff as a cap, but after some current flow it starts brekaing down inside and becoming more resistive.

After all that, the time-delay for the voltage to drop could also be a faulting chip on board, breaking down internally after initial warmup/current flow starts breaking it down.
This will be harder to find, as you literally need to disconnect the 5v pin of EVERY chip until the fault goes away.
Firstly, I'd hope the CPU & PIA's would be socketed, along with ram & rom chips, at least pull all those out first and re-check the 5v stability.

Some of the unused(empty) rom sockets might just be full of dust & crap from storage maybe also causing a resistive path on the 5v supply.

my 2c  ;-)
MM