The LXX reading of Amos 4.13a was used by a heretical sect known as the Tropici to argue that the Holy Spirit was a created being of God. The LXX reading could sound like a Trinitarian reference which would make the Spirit, in this context, not one with Father and Son, but somehow subordinate. In the LXX it reads διότι ἰδοὺ ἐγὼ στερεῶν βροντὴν καὶ κτίζων πνεῦμα καὶ ἀπαγγέλλων εἰς ἀνθρώπους τὸν χριστὸν αὐτοῦ (“For, behold, I am he that strengthens the thunder, and creates spirit/wind, and proclaims to man his Christ”). The MT reads differently. It says וּמַגִּ֤יד לְאָדָם֙ מַה־שֵּׂחֹ֔ו (“For, behold, he who forms the mountains, and creates spirit/wind, and declares to man what is his thoughts”). I am not sure how the translators turned מַה־שֵּׂחֹ֔ו(“his thoughts”) into τὸν χριστὸν (“his Christ”).
The Tropici used this translation to prove the Spirit was a created being because this text mentions God (speaking), Messiah (the Son), and Spirit (which is created). While there are plenty of ways to provide rebuttal to the Tropici one point of interest is that “Messiah” shouldn’t even be in the text it seems. Therefore this text is not speaking of “Father, Son, and Spirit” at all.
Thus far my hypothesis is that this is either a case of homoeoteleuton or homoeoarchton (thanks to Brian Lilly for the definitions). I say this because my assumption is that מַה־שֵּׂחֹ֔ו must have been mistaken for מָשִׁיחַ (“Messiah”). This would make the most sense it seems since if the ה was overlooked it would look like “his Messiah”. My question is does one of the two textual criticism definitions apply or is this a case of something else?
the archives near Emmaus 
Brian, your observation is completely solid. We really don’t know whether LXX was translated based on earlier version of MT (pre-MT) or not. LXX for sure is a lot older that MT that we have now. It seems to me that they are two different textual traditions. If we read the text carefully, many times LXX is not consistently parallel to MT.
In my study about Galilee and archaeology, I found that Galilee in LXX (Joel 4:4) is called γαλιλαία ἀλλοφύλων (foreign Galilee), which is very different from the MT גְּלִילוֹת פְּלָשֶׁת (regions of Philistine). This is the reason why many scholars (especially text based scholars – Martin Hengel, etc.) think that the people who lived in Galilee were Gentiles and not Jews. However, this view has been rejected by many modern archaeologists (Eric Meyer, Jonathan Reed, etc.), that Galilee was thoroughly Jewish.