shirou: Shirou (or Shiro) as you can guess is a Japanese name, relatively common and it means "Fourth (shi) Son (ro)". I suppose there's no particolar reason behind this choice, since we don't really know if his parents gave him this name because he really was their fourth son (but I doubt it, although we don't have detailed info about his original family) or if they just liked it.

emiya: well also Emiya could be used as a first name but it's feminine and I guess this is not our case. Anyway it's a very common one too, if used as a surname, so there's nothing that particular to tell about it.

archer: as you already know, Archer isn't a name that identifies a person, but a category of individuals, or well those soldiers who are equipped with bows (it doesn't matter if on horse or not). The word comes from French and it's a derivation of Arc which is the correspondent word for bow.