Azmi is a masculine name that can also be used for girls, but originally it is a masculine word. The feminine version is Azmiah. Azmi is the attributive form of Azm, which means "determination", "resolution", "resolve", "strength of will".
An attributive form is similar to the way in English "beauty" becomes "beautiful"; Azm becomes Azmi, thus Azmi means "one who has determination", "one who has strength of will". The word Azm is mentioned five times in the Quran, and Azmi is derived from it, thus Azmi is an indirect Quranic name.
There are no word that can be joined with Azmi to create a new meaning. But if you use Azmi as a second or last name, it becomes a description of the name that comes before it. Thus Muhammad Azmi means "Muhammad, the man who has a strong will".
In addition to this, some people like to use a second name that comes from the same root as the first name, in this case Azmi Azzam or Azzam Azmi. Azzam also comes from Azm and has the same meaning as Azmi.