I have Redhat 7.3 with latest kernel (upgraded with rhn-update) in IBM (type 6587LBV, system board 54JF9H9671H, these numbers were get from BIOS information pages) Pentium 200MHz with 64MB RAM. A few times I noticed that system was partially hanged at some time. All local services are unavailable. All services ports (http at tcp 80 etc) are answering but there is no data is coming. Locally connected monitor is dark (energy saving functions were turned off before), LCD display (lcdproc software is showing some information there) is connected to printer port and information there is stopped at last state. Machine has two network cards and is doing NAT for internal network. All users at internal network can access internet well at such hang situation, linux machine even answers to ping requests and all port forwardings from outer network to internal were in good condition, but SSH and ipv6 tunnels are not working. After rebooting machine all works fine until next time. There is no any errors in system logs (no hard disk related errors etc). Also there is no any data coming to other linux machine logging system at such hang time (remote syslogging is configured and working).
And for last, today was same situation, then I tried to access any service, that was configured to work here. Telnet window was just waiting for data. And exactly 30 minutes, after I noticed that system was freezed, about all services starts to work again (noticed "login:" text at telnet window), no need to reboot machine. Even information in LCD display (connected to printer port) was running again without need to restart this service. Some services were dead in linux machine at this 30 minutes freeze time, but almost everything was working again, no any errors in system logs about that 30 minutes time, all was just paused.
My question is: where to start to search and resolve the problem?

Well, first off, I'd look at your memory usage. 64 megs seems pretty small unless it's only acting as a router for a few people and you're not caching too much information. Second, is it cooling properly? Check the fans, heatsinks, and make sure it's clean inside the case. Also, do a good memory scan to make sure you don't have a faulty memory module. Make sure there aren't things running in the background that are sucking up system resources. Of course, do a virus scan and check that somebody else doesn't "own" your box, and make sure the services were properly configured in the first place. (Never assume they were, no matter who configured it.) Lastly, make sure you have enough swap space.

As for the "dark" monitor, has it ever worked? While a broken monitor might not kill the system, there's a small chance that your system can lock up if it's not working right. If you have accelerated drivers (for a server?) it's more likely to happen.

Because it seems to work fine when rebooted, and seems intermittant, I'd guess heating or memory problems.

Although the system is slow, you might run the top command if you can and see if there are any obvious processes that are consuming CPU. This will also give you the info on memory and swap usage.