The simple answer here is probably best: "Son of Abraham, son of David" would be understood by Jewish readers/hearers to mean "a Jew in the line of David, and therefore qualified to be the Messiah." Some additional considerations:
Messianic expectations had been brewing for centuries under various regimes since the Jews returned to Judea from Babylon. The last hope for a Davidic king had been placed in the person of Zerubbabel (Haggai 2:23), who was the grandson of a Davidic king. When Judea finally became independent under the Hasmoneans, however, they established a royal line that did descend from David. For many Jews this was disqualifying and did not fulfill their messianic hope. Matters became worse under the Herodian rulers since they ruled on behalf of Rome and were ethnic Idumeans (Edomites) who had converted to Judaism rather than being lineal Jews. Although the Edomites were technically "sons of Abraham" too, there is no indication historically that they claimed this title (as Muslims do today).
The term "son of Abraham" in the OP also indicates "Israelite." It is used in a similar way to Paul in Romans 11:1.
I myself am an Israelite, a descendant of Abraham, a member of the
tribe of Benjamin.
Similarly, in Acts 22:3, when Paul wished to establish his credentials as a Pharisee he is quoted as saying:
I am a Jew born in Tarsus in Cilicia but brought up in this city at
the feet of Gamaliel, educated strictly according to our ancestral
law, being zealous for God, just as all of you are today.
So, Paul established his bona fides by identifying himself as a son of Abraham (a Jew) who was also a member of the tribe of Benjamin: someone who took his Jewish ancestry seriously. In Acts, he is both Jew and Pharisee in the spiritual lineage of the famous teacher Gamaliel, thus able to give well-founded rabbinic opinions. Similarly, Matthew established Jesus' messianic credentials by showing him to be a descendant of Abraham who was in the line of King David, the progenitor of the Messiah.
The phrase "son of Abraham, son of David" thus qualified Jesus on two important bases that were missing from recent occupants of the Jewish throne. The Hasmoneans were not descended from David, and the Herodians were not even ethnic Jews.