None of the six Hebrew words for 'restore' are used in Amos 9:11-15 according to Young's exhaustive concordance (see pages 811-2). It is the Hebrew word 'qum' which means 'To cause to rise up'.
Young renders Amos 9:11 as "In that day will I raise up the tabernacle of David... I will raise up his ruins, and I will build it as in the days of old." (K.J.) Amos is saying that God is going to cause to rise up the tabernacle, and the ruins of David's inheritance, which the people in Amos's days had so squandered that God was going to adversely judge the people.
The surrounding context (which you asked about) helps give the meaning. Up till then Amos had been uttering so many words of condemnation to God's people, it seems surprising that he should end his prophecy with a wonderful picture of future blessings. David is the kingly figure (which is why "his" ruins is the right rendition, not "the" ruins). It is the Son of David promised in 2 Samuel 7 that is the focus of Amos's thinking.
The broken places Amos speaks of is a reference to the division between the north and the south.. So, when God says he will "restore his ruins", this refers to David, the ruin of the kingly line, an almost exclusively Messianic prediction that the Lord will raise up from David this ruler who will come and build the Davidic dynasty as it used to be. In the future, even those enemy nations, such as Edom etc, will be incorporated into a time of blessings to come after God has punished his people.