What's the word for something that has a mind of its own?

I like the expression "self-aware"

The computer which controlled the machines, Skynet, sent two Terminators back through time. .... It becomes self-aware at 2:14 a.m. Eastern time, August 29th.

Usually, when something - mechanical/electronic/non-human animal- is described as 'having a mind of its own', it's because it doesn't agree with the person describing it. In those cases, the thing is often called by the personality perceived. Ex:

"An ornery old truck."

"A moody computer."

"A snobbish goldfish."

When something (like, my car) exhibits a multi-faceted personality, I am forced to concede that she has a soul.

Besides Human intelligence, we have - Artificial intelligence: (AI)

Artificial general intelligence (AGI) is the intelligence of a (hypothetical) machine that could successfully perform any intellectual task that a human being can.

The intelligence exhibited by machines, (lawnmower etc.) or software (watsapp etc.)

(wiki)

Nonsentient (mechanical/electrical) devices that don't always work as intended are often called...

temperamental (informal) working erratically and inconsistently; unreliable
a temperamental sewing machine

There are even a few written references in Google Books to a temperamental lawnmower - but as you might expect, a temperamental computer is far more common.

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If you want to more explicitly "anthropomorphize" (assign human characteristics to) the device, you might say it's got a mind of its own.