Of course we can't be sure why Matthew chose to refer to someone as "the mother of Zebedee’s sons," but we can furnish a reasonable hypothesis by examining the evidence.
The first piece to the puzzle, is to determine who this woman was exactly. This question is actually rather simple. Later in Matthew, we find:
There were also many women there, looking on from a distance, who had followed Jesus from Galilee, ministering to him, among whom were Mary Magdalene and Mary the mother of James and Joseph and the mother of the sons of Zebedee. (Matthew 27:55-56, ESV)
Ἦσαν δὲ ἐκεῖ γυναῖκες πολλαὶ ἀπὸ μακρόθεν θεωροῦσαι, αἵτινες ἠκολούθησαν τῷ Ἰησοῦ ἀπὸ τῆς Γαλιλαίας διακονοῦσαι αὐτῷ· ἐν αἷς ἦν Μαρία ἡ Μαγδαληνή, καὶ Μαρία ἡ τοῦ Ἰακώβου καὶ Ἰωσὴφ μήτηρ, καὶ ἡ μήτηρ τῶν υἱῶν Ζεβεδαίου (Nestle 1904)
while Mark has in a parallel passage:
There were also women looking on from a distance, among whom were Mary Magdalene, and Mary the mother of James the younger and of Joses, and Salome. (Mark 15:40)
Ἦσαν δὲ καὶ γυναῖκες ἀπὸ μακρόθεν θεωροῦσαι, ἐν αἷς καὶ Μαρία ἡ Μαγδαληνὴ καὶ Μαρία ἡ Ἰακώβου τοῦ μικροῦ καὶ Ἰωσῆτος μήτηρ καὶ Σαλώμη,
The high degree of textual similarity here (especially clear in the Greek) virtually guarantees these two passage represent a literary dependency. I will come back to the nature of the dependency; for now, it is sufficient to note that the two authors are describing the same woman. Mark calls her "Salome" and Matthew calls her "the mother of the sons of Zebedee." Thus, throughout Christian history Salome has been identified as the mother of two of Jesus disciples - James and John, who elsewhere are identified as "the sons of Zebedee" (e.g. Mark 10:35).
More on Salome
Before returning to the core question, it might be helpful to briefly outline who Salome was. In John's account of the crucifixion, he records the women at the scene as:
his mother and his mother’s sister, Mary the wife of Clopas, and Mary Magdalene. (John 19:25)
on this basis, some commentators conclude that Salome was Mary's sister. Unlike the connection between Mark and Matthew, however, this conclusion is not solid. "His mother's sister" could indeed refer to Salome, but it could also refer one of the unnamed women of Mark/Matthew.
Elsewhere, Mark 16:1 places Salome at the tomb when it is discovered empty, along with Jesus' mother and Mary, mother of James and Joseph. Matthew's parallel account lists only the two Marys.
In later, Gnostic writings Salome appears several times. In the Gospel of Thomas she declares herself a disciple of Jesus. A preserved fragment of the Gospel of the Egyptians calls her a disciple and implies she is unmarried and childless. In the Pistis Sophia, Salome takes on an importance well below Mary Magdelene (who was very important in Gnostic writings), but on par with an "average" Male disciple. An expanded version of Mark (the so-called "Secret Mark"), added at least one additional verse about Salome during Jesus' ministry. The more orthodox Protevangelion of James, an infancy narrative, makes Salome Mary's midwife and the first believer in Jesus' mission.
Salome's prominence in the Gnostic writings is surprising given that she appears by name only in Mark's gospel, the least popular of the Gospels in the early church. To explain this prominence, scholars have suggested there probably was an early Salome tradition, independent of the Gospels. While the later works she appears in generally have no historical value, we can be fairly confident that, at minimum Salome was a real person and a follower of Jesus during his lifetime. There is also a decent chance she played a notable role in the early Church.1
Returning to the question of the relationship between Mark and Matthew, we have three potential data points:
1) Mark says "Mary Magdalene, Mary the mother of James, and Salome bought spices so that they might go to anoint Jesus' body" (16:1) while Matthew says "Mary Magdalene and the other Mary went to see the tomb" (28:1).
- There is little verbal similar here, so the two accounts probably represent independent accounts of the same event.
2) The passage that spurred the OP's question. In Matthew (20:20-28), "the mother of the sons of Zebedee" asks Jesus to place her sons at his right and left hands when his kingdom comes. In Mark's account (10:35-45), James and John ask directly.
- These passages certainly have a dependency. The language used is quite similar and, despite their dissimilar openings, both accounts have Jesus' reply directed at James and John.
3) The lists of women at the crucifixion, mentioned above (Mark 15:40, Matthew 27:55-56)
- As previous stated, the accounts show dependency.
So who depends on who? That depends on what one decides in the redactional activity here.
If one is committed to absolute Markan priority, then the differences must be from the redactional activity of Matthew. In this view, Matthew must have had a reason to change the requester in Mark 10:35-45. Perhaps he felt it embarrassing for two great disciples to make such a request, so put it in their mother's mouth instead.
In favor of this view, it does supply a satisfactory reason for the phrase "mother of the sons of Zebedee" as the passage is really about "the sons of Zebedee," as James and John are often referred to collectively, so identifying the speaker as their mother instead of by name makes sense. Against this view, Matthew is not afraid to have disciples make unreasonable requests/say stupid things elsewhere, so "inventing" a speaker here to buffer the question does not seem to something typical of Matthew's redaction. Furthermore, does it really make James & John look any better? They are still portrayed as wanting the glory/not understanding Jesus' mission/kingdom properly, but now have the additional potential embarrassment of not even being bold enough to ask themselves, instead making their mother do it for them.
The greater challenge for a Markan priority view, though is Matthew 27:55-56. Here Matthew allegedly changed the straight forward "Salome" for "mother of the sons of Zebedee". If his purpose was to connect the women to the disciples, why not use "Salome, mother of ...", similar to how he describes the second Mary, instead of just "mother of ..."? Gundry can only offer the weak (in my mind) explanation:2
The obscurity of Salome, the prominence of the sons of Zebedee, and the desire for a parallel with the immediately preceding description of Mary as a mother (whose prominence will not allow the loss of her name) all influence Matthew's revision.
All of these things could indeed push Matthew to add "mother of...", but one struggles to see how any would necessitate also dropping "Salome". The only half explanation - that Mary was supposedly more prominent - is pure speculation and ignores that "Salome, mother of..." is a stronger parallel to "Mary, mother of..." than simply "mother of..." is, if, indeed, Matthew wanted to make a verbal parallel.
It is reasonable to assume Matthew wanted to connect the women at the crucifixion to his description of her earlier in his Gospel, but that is an equally valid assumption under any dependency theory. If one is not committed to Markan priority, an overall easier explanation arises: Mark is simplifying Matthew on the two dependent passages. This is equally valid under an absolute Matthean priority theory and a more flexible theory that allows for both Gospels to preserve the more primitive tradition at some points.
In this view, Mark sees that James and John are the real focus of Matthew 20:20-28, and understands that the request, although spoken by Salome, really comes from James and John and thus puts it in their mouth (or believes that the request was made collectively). He thus simplifies the passage by omitting Salome's part. Likewise, in 15:40 he simplifies "the mother of Zebedee's sons" because he knows her given name and see no reason for the complicated formulation.
Perhaps Mark had good access to the "early Salome tradition" mentioned above and thus knew well who she was. This hypothesis also has the benefit of explaining the anointing verse (16:1), which is the only resurrection passage in any of the four Gospels that mentions Salome. If Mark had superior knowledge about Salome, it would explain how he knew this additional detail, which might have been unknown to Matthew.
Why "Mother of Zebedee's sons"?
The question then is, why does Matthew refer to Salome as the "Mother of Zebedee's sons" instead of "Salome" or "wife of Zebedee"? We can quickly suggest a good reason why she would not be referred to as "wife of Zebedee" -Zebedee is not an important figure in the Gospels. There is no indication he ever became a follower of Jesus and the only time the man appears is when James and John leave him in a fishing boat to go and follow Jesus.
Salome's sons, of course, are important figures in the Gospels. Thus, it is obvious that "mother of..." is a clearer/better reference than "wife of..." (A few commentators add a possibility that Zebedee has died by the time Salome was so referenced, but I find this to be an unnecessary assumption and insufficient explanation.)
As to why Matthew uses "mother of..." instead of "Salome," there are two plausible explanations. The first is that he wants to emphasize her connection to James and John. This formulation especially makes sense in 20:20-28 where her role in the story is unimportant. It would indeed be natural to say something like "John's mom came and made a demand for him" in a situation like what is described, even if one knew John's mom's name. Since James and John are often collectively referred to as the "sons of Zebedee", there is no reason why "the mother of sons of Zebedee" would have any different connotation than "the mother of James and John". It is possible, then, that Matthew simply repeated the same formulation in his crucifixion account to make it clear he was referring to the same woman. This explanation works perfectly under Matthean priority, and adequately under Markan priority (where it is weird, but plausible that "Salome" fell out).
The second explanation is that Matthew simply did not know Salome's given name. This would not be impossible even if the author of Matthew is the apostle Matthew - he would not necessarily know/remember the name of a woman he met only a couple times. Still, it would seem easier to support this hypothesis if the apostle was not the Gospel's author. This would make it harder to see Matthean priority, but not implausible if some followers of Matthew wrote the gospel down after his death, using his eyewitness testimony which did not happen to include Salome's name.
In light of the evidence, the simplest/best explanation is that Matthew wrote 20:20-28 and 27:55-56 before Mark wrote his parallel passages (or more precisely that Matthew's versions of the passages is the more primitive tradition). He used "mother of Zebedee's sons" in 20:20 because the story was primary about Zebedee's sons, and thus it was completely natural to describe the speaker as their mother. He then carried that formulation forward to his crucifixion account. Matthew may have known Salome's name, but didn't see a need to mention it if he did.
Mark 10:35-45 then represents a simplification of Matthew's account. Consequently, in the crucifixion account, he had no use for the long formulation "mother of... " and so substituted Salome's actual name. He quite possibly also had access to independent information about Salome, which explains the fact that only he mentioned her in his empty tomb account.
Finally, I agree with Jonathan Chell's suggestion that Matthew's formulation allows an otherwise unidentifiable women to be associated with her very important sons. Divine intervention, of course, cannot be proven as a hermeneutical explanation, but it certainly is interesting that without Matthew's "odd" phrasing Salome would not have received the credit (by name) she deserves for raising the great apostles of James and John.
1 Extra-biblical information on Salome comes mostly from Gospel Women: Studies of the Named Women in the Gospels by Richard Buckham
2 Matthew: A Commentary on His Handbook for a Mixed Church Under Persecution by Robert Horton Gundry