Khan is an Mongolian title origin if I recall. Mirza comes from Arabic. Does Mirza not derive from Amirzade? Amir in Arabic means prince. Don't know if there is a connection there but it is very probable.
But yes, surnames don't say that much indeed. But they can give an indication. It depends where in the world. Many Danes for instance took German names, especially those from the cities because it was considered prestigious but most of them don't actually have ties to Germany despite the name. But in Iran, which borders the Arab world and has been tied to it for nearly 1500 years and where Arabs form one of the biggest ethnic groups and where there have been a lot of migrations it is not that unusual if you have people being partially Arab. Especially in regions of Iran where Arabs are known to live.
Iranian Arabs - Wikipedia, the free encyclopedia
This Wikipedia claims that those are Arabs (the avatars) but by their names one would never have guessed this. So I guess this goes both ways.
Where is your mother from in Iran if I may ask @
SinaG ?