Symptoms
Random reset, sometimes resetting rapidly

Solution Steps
Looking at the schematics, reset is controlled using a Power On Reset (POR) IC. Basically how it works is that it sends a reset pulse to the CPU whenever a threshold voltage is reached, somewhere in the 4V range. With that knowledge, I had to monitor the reset line.

In this image, the reset line is yellow and the +5V rail is in blue. What I wanted to do was see if there was maybe a short or if there was excessive noise going on that went below the threshold. To my surprise, it periodically dipped a lot lower than expected, causing the reset. We definitely have a power supply issue at this point.

Looking at the power supply circuit for +5V, there isn't a whole lot going on here. How it works, is that an AC line from set of taps directly from the transformer gets rectified and filtered. This makes a unregulated +/- 12V DC. From there, the +12V DC circuit is regulated to 5V using a 2N6057 and MC1723 circuit. 

Looking at it, my initial guess was the rectifier as these fails often. So I test it in circuit, and suprisingly one of the diodes in it was shorted, specifically on the ones creating the +12VDC line. So if it was failing, it could explain the dip in 5V. What was super interesting was that it didn't pop the fuse. Normally when one of these fails, they take out the fuse. Replacing the rectifier fixed the issue and it hasn't improperly reset since I replaced it with a new one.

Solution
Bad power supply bridge rectifier at DB1.