As I wrote here four months ago the Greek root meaning “same” is homoios (ὅμοιος) or “homos.” From the first mainly words starting with “homeo-” are derived.
The Latin word “homo” means “human being” and does not imply any gender. “Man” (male human being) in Latin means “vir,” “woman” means “femina.”
By the way, since Zamenhof took many roots from French (then the predominant language) he might also have been influenced by the French word “homme.”
Not a translation, but PIV defines it ( http://vortaro.net/#peni ):
Streĉi la korpajn aŭ mensajn fortojn, por atingi rezulton
So, something like "make the effort to, do one's best to, struggle, try" or something along those lines. See also the examples/quotes there (many from Zamenhof) to get a feel for that word.
It's likely the difference between "human" and "person", human refers directly to our species, whereas person is less of a solid concept, and can generally refer to any individual with a personality - there are those who say "dogs are people too!" (or insert any non-human animal really), which you are free to disagree with and debate on, however few rational people would try to argue that "dogs are humans too!" because they physiologically aren't.
http://esperanto-panorama.net/vortaro/eoen.htm Also, "personoj" can be "persons" if the plural means individuals rather than the whole group of "people". http://www.gutenberg.org/files/16967/16967-h/16967-h.htm "man" can be "viro" or "homo". "person" is "persono" "people" can be "popolo" or "homoj"
In English, the word man, e.g. the rights of man, is not limited to males. This phrase I take to be an Esperanto translation of Terence's "Homo sum nihil humanum alienum a me ❤❤❤❤," which is traditionally translated "I am a man; I think nothing human foreign to me." It is particularly appropriate for an Esperantist, but sounds quite stilted to me when translated "I am a person."