I would try Windows 2000 or Windows XP and use the system for a week or two - both for regular use and run benchmarks / stress tests from time to time.
My personal experience is that sometimes benchmarks & stress tests won't trigger a faulty hardware to act up. Yet regular "on/off" and intermittent use can. So that's why I suggest to use the system with one of the above OSes to see how that goes. If nothing crops up in a week or two or more in using the system, I'd say it's a Windows 9x issue, possibly either something with the installation itself or the drivers used for it.
As for the PSU, I agree with zb10948 above - 150 Watts should be plenty of power for your system, provided it's a good quality PSU. Most Pentium 3 systems rarely pull more than 40-50 Watts from the wall with a "basic" GPU like GeForce2 MX 420/440/460 or Radeon 7000/7200 (yes, I've measured a lot of my systems and was always surprised how efficient P3 is.) With Pentium II, the power jumps up a bit, but not much - generally somewhere in the 50-60 Watts tops under load. Most of this is from the 5V rail... but even then, that's like 10-12 Amps tops. So if your PSU is capable of providing that much and it's a known decent brand, then it should be fine.
To anyone who thinks you need a 200-300 Watt PSU minimum for a P2/P3/early P4 system: buy yourself a Kill-A-Watt meter or similar equivalent wall power meter (that can measure the real power usage), and you will see how much each of these actually use (hint: not as much as you might think. 😉 )