I would still do a visual inspection of all capacitors as a starting point. These are a fairly obvious and relatively inexpensive fix, although could be tricky to remove and replace depending on your desoldering and soldering skills. (If a leaky cap is on the PC mainboard then it can be difficult to dislodge.)
Could be RAM - if certain areas of the RAM are only used occasionally this could explain the symptoms. But start with heat as the most likely cause of the symptom.
You can always get a hair dryer or heat gun on to different areas to see if the reset occurs (use with caution!!). Get someone else to play while you heat up different areas of the main board, power supplies and then driver boards. This will narrow it down for you, most likely.
