A. Rambam (Melochim 1:4) based on Talmud (Yevamos 45b, Sotta 41a) rules that we may not appoint a king that is a descendant of converts, even after many generations, unless or until his mother is Jewish. The Rambam quotes as his source the verse: (Devarim 17: 15) "You cannot place over yourself a foreign man, who is not your brother" which would indicate that if his mother is Jewish he is called a brother.
However Horav Shlomo Miller Shlit"a pointed out that not all agree (see Hagohos Maimoyi ibid.) quoting another source mentioned in Tosefta, mainly that he must fulfill also the positive commandment of Mikerev Acheicha, meaning the chosen and best of your brothers and even if his mother was Jewish he would not qualify.
Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit'a