Randomization of events in old hand-held games
From the older machines that I have looked at, often a clock was kept running via a small cell battery. And the time on that clock at start up was checked and used for various calculations. I'll try to dig up a concrete example when I get home from work!

Edit:

One common way of implementing this was through the classic 555 timer chip, circia 1971 (Either the IC or designing one with electronic components) An explanation of the chip can be found here:

http://en.wikipedia.org/wiki/555_timer_IC

This chip can typically be found in any hobbyists toolbox, but how do we use this chip to generate random numbers? The short answer is that we cannot, we must use this in conjunction with other coutning circuits etc. While today these can be found in convenient IC packages, they can be recreated with common components. I tried to prepare a good example, but managed to find a much better one online! An example of such can be found/explained here:

http://www.engineersgarage.com/electronic-circuits/random-number-generator-using-7-segment-display

Hope this helps! :)