It seems that randomization can serve as a general substitute for memory. It's not a perfect substitute, but whenever you don't want to, or can't, remember something for some reason, randomization might help.

Besides the example of The Absent-Minded Driver, I've realized there are other examples in cryptography:

• nonces - Many encryption schemes require a unique nonce to be generated for each message to be encrypted. You can either pick random nonces, or keep a counter. But keeping a counter might be too troublesome, and you might be running in a virtual machine that can be rolled back from time to time, so it's usually better to use a random nonce, even though you'll need a longer nonce than if you used a counter (to keep the probability of collision sufficiently small).
• distributed key search - You can either have a central server that hands out regions of the key space to search, or each participant can search a random region. The latter is less efficient in computing time, but more efficient in communications cost.

I might do a post on this, if I could figure out a way to think about why randomization substitutes for memory.