11 “On that day I will raise up The tabernacle of David, which has fallen down, And repair its damages; I will raise up its ruins, And rebuild it as in the days of old; 12 That they may possess the remnant of Edom, And all the Gentiles who are called by My name,” Says the LORD who does this thing. Amos 9:11–12 (NKJV) 

11 On that day I will raise (anhistemi) up the tent (skene) of David that has fallen (pipto), and I will rebuild (anoikodemeo) its things that have fallen (pipto), and I will raise (anhistemi) up its things that have been destroyed (katascapto), and I will rebuild (anoikodomeo) it just as the days of eternity (aion), 12 so that the remnant (kataloipos) of the people (anthropos), and all the nations (ethnos) upon whom my name was invoked (epikaleo) upon them, will search (ekzeteo) for me,” says the Lord who is making (poieo) these things. Amos 9:11–12 (LES2) 

The main differences between the MT and LXX in this passage are the following:

1. The MT says the tabernacle or tent of David will be rebuilt “as in the days of old,” but the LXX says, “as the days of eternity. Perhaps this is to signify that the church has been in God’s heart from eternity. (Ephesians 1:4)
2. The MT says “they” will “possess the remnant of Edom and all the Gentiles who are called by my name.” The Hebrew word means to “possess or dispossess, especially by force.” This is what the nation of Israel is currently doing in Gaza, Lebanon, etc. The LXX, however, turns this completely around by saying the “people and all the nations…will search for me.” As is often the case, the MT elevates Israel, but the LXX elevates Christ and the church. It is interesting to read commentary based on the MT. Bible scholars work hard at trying to make sense of what I consider to be misleading text..

James, the Lord’s brother, the overseer of the church in Jerusalem quoted this passage at a conference discussing how the law of Moses would be applied to the Gentiles.

13 And after they had become silent, James answered, saying, “Men and brethren, listen to me: 14 Simon has declared how God at the first visited the Gentiles to take out of them a people for His name. 15 And with this the words of the prophets agree, just as it is written: 16 ‘After this I will return And will rebuild (anoikodemeo) the tabernacle (skene) of David, which has fallen down (pipto); I will rebuild (anoikodemeo) its ruins (katascapto), And I will set it up (anorthoo); 17 So that the rest (kataloipos) of mankind (anthropos) may seek (ekzeteo) the LORD, Even all the Gentiles (ethnos) who are called (epikaleo) by My name, Says the LORD who does (poieo) all these things.’ 18 “Known to God from eternity are all His works. Acts 15:13–18 (NKJV) 

By carefully comparing the LXX to the NT quote, we see that almost all the words in Greek match. One difference is the verb in the LXX for “rebuild” is “anoikodemeo,” but the NT uses “anorthoo.” James omitted the phrase, “just as in the days of eternity.” As for other minor differences, we need to remember that James was citing the OT text from memory rather than reading from the LXX.  James also omits adding “all the nations” after saying that the remnant of mankind will seek the Lord.

Taking all of this together, it is clear that James quoted the LXX. It is also clear that the LXX points to Christ and the mystery of the Gentiles’ inclusion in God’s kingdom.