When a person is described as a Davidide, what is being said about that person? In my NABRE Bible, there are footnotes on Isaiah 11:1-9 (10) that say Isaiah is "looking forward to a new Davidide who will realize the ancient ideals (see Psalm 72). The oracle does not seem to have a particular historical person in mind." I think it might be one of David's descendants but I am not sure as I have never come across or heard this term. Psalm 72 is a prayer for the king.
Isaiah chapter 11 is about the heir to David’s throne.  It is a Messianic prophecy.  Psalm 72 reflects on the prospects of David’s royal line and on Zion.  The surpassing righteousness and domination sought in this prayer foreshadow the coming of Jesus, the Son of David.  Source: New Living Translation Study Bible notes.

Isaiah 11:1-10 describes the paradise of the Messiah’s triumph.  The reference to Jesse, the father of David indicates that this continues the theme about the coming heir of David (the Messiah).  Psalm 72 is a royal psalm, praying that the heirs of David’s line (beginning with Solomon) might have success...  Source: English Standard Version Study Bible notes.

Although the word ‘Davidide’ is not found in any of my Bibles, it apparently refers to a descendant of David:


  Ezra 8: “family accompanying Ezra, Hattush, is a descendant of David (so called "Davidide"), and he would be the fourth generation after Zerubbabel (cf. 1 Chronicles)...
  
  But this united monarchy—a real united monarchy—was ruled by the Omrides, not the Davidides, and its capital was Samaria, not Jerusalem.  https://en.wikipedia.org/wiki/Historicity_of_the_Bible


As best I can establish, the word ‘Davidide’ means “son of David”:


  Not all Davids look alike: Dave, Davey, and Davi; even Dudel, Daw, Taavi and Taffy are etymologically Davidide (Yes, this actually is a word. It means “son of David”). Not to be gender exclusive, David comes in feminine forms like Davida and Davena.[1]  http://www.behindthename.com/name/david