It cannot mean “the first Sabbath” because the lemma σάββατον is a noun declined in the neuter gender, while the cardinal number μία (an adjective) is declined in the feminine gender. In Greek grammar, an adjective must agree in case, gender, and number with the noun it modifies.
If we are to suppose that τῇ μιᾷ τῶν σαββάτων is to be translated as “the first [Sabbath] of the Sabbaths,” with the supplied ellipsis “Sabbath,” then the cardinal number should be declined in the neuter gender in agreement with supplied ellipsis τῷ σαββάτῳ, as in τῷ σαββάτῳ τῷ ἑνὶ τῶν σαββάτων—“the first Sabbath of the Sabbaths.”
The fact that the cardinal number is μία, declined in the feminine gender, suggests that the supplied ellipsis should be ἡμέρᾳ (“day”), as in “the first [day] of the week,” since ἡμέρᾳ is also declined in the feminine gender.
In English, it is also common to use such ellipses when numbering days. For example, in the phrase “It is the first of the month,” the word “day” is omitted because it is assumed to be understood by the reader.
Unlike Hebrew, English actually has unique names for the days of the week, such as Monday, Tuesday, etc., which of course are derived from the names of astronomical phenomena (e.g., Sunday, Monday) or gods (e.g., Thursday, Saturday). Therefore, it is unnecessary in English to refer to days of the week by number.1 On the other hand, it is typical to do so in Hebrew.
For example, the following chart displays how the days of the week were numbered in the Talmud:

It is noteworthy that בשבא and בשבתא/בשבת are equated in such phrases as חד בשבא and חד בשבתא, both evidently meaning “the first day of the week.”
In addition to the aforementioned phrases, there also exists the phrases כל השבת and באמצע שבת, meaning “the whole week”2 and “in the middle of the week,”3 respectively.
The Greek Septuagint, written a few centuries prior to the Christian advent, has the phrase τῆς μιᾶς σαββάτων in Psa. 24:1, a psalm that was historically sung on the first day of the week by Levites in the Temple, as testified by Jewish writings.4

This is impartial evidence identifying τῆς μιᾶς σαββάτων with the first day of the week, as Collins notes,5
...because we have good evidence that Jewish Greek used μία for the first day of the week, in the expression ἡ μία [ἡμέρα] τῶν σαββάτων “the first [day] of the week.” For example, in the LXX inscription to Ps 24 (lxx Ps 23.1), we find:
ψαλμὸς τῷ Δαυιδ τῆς μιᾶς σαββάτων
A psalm for/of David, of the first day of the week.
In addition to the preceding arguments against the translation “the first [Sabbath] of the Sabbaths,” a few points could benefit from clarification.
It is argued by some that σάββατον should never be translated as “week,” for the author would have used ἑβδομάς instead. Both in the Old and New Testament, there are instances where the author is clearly referring to a “week” rather than “Sabbath” using the word σάββατον (or a declension thereof).
- Luke 18:12: νηστεύω δὶς τοῦ σαββάτου (“I fast twice a week”)
Regarding this verse, Francis D. Nichol wrote,6

In addition, there are the following verses in the Greek Septuagint:
- Psa. 24:1: τῆς μιᾶς σαββάτων (“the first day of the week”)
- Psa. 94:1: τετράδι σαββάτων (“the fourth day of the week”)
Historical sources testify that both of these psalms were sung in the Temple by the Levites on their respective day of the week.
Furthermore, in Lev. 23:15, it states, שֶׁבַע שַׁבָּתוֹת תְּמִימֹת תִּהְיֶינָה—“and seven שַׁבָּתוֹת shall be complete.” Some translate שַׁבָּתוֹת as “Sabbaths,” as though the Israelites had to count seven complete Sabbaths, but consider the following remarks from Carl Friedrich Keil,7
That שַׁבָּתוֹת (ver. 15) signifies weeks, like שָׁבֻעֹת in Deut. xvi. 9, and τὰ σάββατα in the Gospels (e.g. Matt. xxviii. 1), is evident from the predicate תְּמִימֹת, “complete,” which would be quite unsuitable if Sabbath-days were intended, as a long period might be reckoned by half weeks instead of whole, but certainly not by half Sabbath-days.
References
Collins, C. John. “The Refrain of Genesis 1: A Critical Review of Its Rendering in the English Bible.” Technical Papers for the Bible Translator. Vol. 60, No. 3, July 2009.
Keil, Carl Friedrich. Commentary on the Old Testament. 1900. Reprint. Trans. Martin, James. Vol. 1. Grand Rapids: Eerdmans, 1986.
Nichol, Francis D. Answers to Objections: An Examination of the Major Objections Raised Against the Teachings of Seventh-Day Adventists. Fort Oglethorpe: TEACH Services, 2014.
Phillips, John. Exploring Psalms: An Expository Commentary, Volume One. Grand Rapids: Kregel, 1988.
Footnotes
1 Thus, it is typical to say, “Today is Wednesday” rather than “Today is the fourth of the week.”
2 Chul. 26b; Meg. 30a; Men. 65a; Nazir 5a; Yoma 46a
3 Taʿan. 29b, 30a, etc.
4 See Phillips, p. 180; also, cp. Babylonian Talmud, Seder Moʿed, Tractate Rosh ha-Shana, Chapter 4, Folio 31a, Gemara (English translation)
5 p. 124–125
6 p. 237–238
7 p. 442