Well, there is some randomness in it. It mostly depends on a flow of the sentence. When 'him' is the first word in a sentence, it never is HO. Just does not sound right. While if you use a pronoun JÁ here, it would change to Já ho nevidím. Simply because Já jeho nevidím is a mouthful. Yet some people actually would say that as well. It would point out that they see other people but not him. So, to a certain point, it is a hit and miss type of a word.
I believe this is also a valid linguistic reason. The same thing with personal pronouns occurs in my mother tongue as well. Also, there is something called "Principle of economy" in language, the fact that languages follow one overarching principle - "the principle of least effort". We feel good and it sounds cool when we say a lot in a few words :) Soundy-feely is also linguistics!
In a normal sentence: "Nevidím ho", the pronoun can refer to any masculine noun, so it can be "it", indeed (Kde je ten pavouk? Nevidím ho! - Where is that spider? I don't see it!)
But the pronoun is stressed here ("HIM I don't see, although I see someone else") and even though I'm trying, I can't quite imagine using it for anything else than a person. In my spider example, I'd just include the noun instead: "Toho pavouka nevidím!")