Do サトシ (Satoshi) or ナカモト (Nakamoto) mean anything in Japanese? Do サトシ (Satoshi) and ナカモト (Nakamoto) have any special meaning in Japanese?

For example, the last name "Smith" or "Miller" refers to a blacksmith or grain miller, respectively.
Without knowing the intended kanji, it's unclear exactly what meaning "Satoshi" and "Nakamoto" have, because there are sometimes many homophones in Japanese (including homophonous names). The Japanese Wikipedia article on Satoshi Nakamoto does not give kanji (and also treats it, typographically, as a Western name), suggesting that the exact intended meaning is unknown. But, we can make educated guesses.

For "Satoshi", there are over a dozen possibilities listed on a Japanese name site, but many of them relate to virtue, wisdom, intelligence, and other intellectual traits.

For "Nakamoto", it's much easier to provide a solid guess. 中本 is a common family name, roughly meaning "middle-origin". 仲本 might also be plausible, with this "naka" meaning "relation, relationship". And, even if "moto" were written as 元, this still roughly means "origin".