Chop it up and swap it around. See if the meaning is still there. "Anything he said, I didn't pay attention to it." The point is, you didn't pay attention. I'd really have to stretch my imagination to accept B as a possibility. The basic idea of the sentence is that you were ignoring him. A double-meaning doesn't jump out here. The only alternative reading I can think of is: "I didn't pay attention to any thing he said." 'Any thing' meaning 'any particular thing'. So you paid attention, but not what he said. Maybe the tone of his voice, or that he needs a shave, or that the sky was very blue that day... so you got some kind of impression, but little to do with his words.

"I didn't pay attention to anything he said." only means: "I ignored everything he said"

I would say that this sentence has one meaning, which you clearly outline in option A. Option B, "I didn't pay attention to just anything he said", implies that the subject was being selective in what they paid attention to -- that certain parts of what was said were of greater value to the listener. This differs in meaning from your original sentence, because the subject has paid attention to some of what was said.