It wasn't extra capacity so much as making some of the internal tools publicly available, and it took off.

I believe SQS was the first publicly available service. EC2 was obviously the tipping point and that was a couple of years later.

The story I always heard was they had a ton of extra capacity that was very seasonal and then someone came up with the great idea to sell that extra capacity in the off season.

SQS as the first publicly available service sounds weird to me but I don't have any sources to cite otherwise.

Edit: Wikipedia says SQS was the first service https://en.wikipedia.org/wiki/Amazon_Web_Services#History

I recall S3 being the first product announcement, with no real mention of what was yet to come.

SQS came before S3.