In 1 Tim. 4:10, the Greek text states,
εἰς τοῦτο γὰρ καὶ κοπιῶμεν καὶ ὀνειδιζόμεθα, ὅτι ἠλπίκαμεν ἐπὶ θεῷ ζῶντι ὅς ἐστιν σωτὴρ πάντων ἀνθρώπων μάλιστα πιστῶν TR, 1550
which may be translated as,
because for this reason we both suffer and endure reproach, since we trust the living God, who is the savior of all men, especially faithful.
On the Greek Word μάλιστα
The Greek word μάλιστα can certainly be translated as "especially" or "particularly," as affirmed by BDAG and Thayer.
On the Soteriology of 1 Tim. 4:10
For those who believe that all men shall be saved without qualification (i.e., "universal salvation"), why would the apostle Paul say "especially faithful"? It would seem that if salvation applied to all, equally and without qualification, then the disclaimer "especially faithful" would have been unnecessary.
While it is true that God "wants all men to be saved,"3 this refers to the antecedent will4 of God, whereas the consequent will5 requires the just God to punish those so deserving.6 According to the New Testament, those who are in Christ are righteous and are saved from divine punishment, whereas those who do not believe in Christ are unrighteous and suffer God's wrath.
While salvation is a gift gratuitously offered to humanity by God, the gift must be accepted by believing in His Son,7 the Lord Jesus Christ, the gift of God.8 Thus, objectively, God is the savior of all men, through his Son, the Lord Jesus Christ, but this salvation must be subjectively actualized by believing in the Lord Jesus Christ. Hence, God is the savior of all people, objectively, but especially of faithful [people], subjectively, for they alone appropriate salvation by believing in the Lord Jesus Christ.
As Henry Alford commented on 1 Tim. 4:10,9
especially them that believe (in these alone does that universal salvation, which God has provided, become actual. He is the same σωτήρ towards and of all: but these alone appropriate His σωτηρία.
August Bisping commented on 1 Tim. 4:10,10
which may be translated as,
God is the savior of all men according to His will; He desires to lead all men to salvation (cp. 1 Tim. 2:4). But this will is itself only realized by the believers who truly obtain the σωτηρία (“salvation”). Thus, for these God is the σωτήρ (“savior”) in the actual and eminent sense.
1 p. 613
2 p. 387
3 1 Tim. 2:4: «πάντας ἀνθρώπους θέλει σωθῆναι»
4 τὸ ἑπομένον θέλημα; voluntas antecedens
5 τὸ προηγουμένον θέλημα; voluntas consequens
6 First remarked by John of Damascus in An Accurate Exposition of the Orthodox Faith, Book 2, Ch. 29, and further expounded by Thomas Aquinas in Summa Theologica, I, Q19, A6, ad 1.
7 cp. Acts 16:30-31
8 Isa. 9:6 cp. John 3:16
9 p. 340-341
10 p. 131
Alford, Henry. The Greek Testament. Vol. 3. Boston: Lee, 1878.
Arndt, William; Bauer, Walter; Danker, Frederick William. A Greek-English Lexicon of the New Testament and Other Early Christian Literature. 3rd ed. Chicago: U of Chicago P, 2000.
Aquinas, Thomas. Summa Theologica. Vol. 1. Paris: Bloud, 1880. (171-173)
Aquinas, Thomas. The Summa Theologica of St. Thomas Aquinas. Trans. Fathers of the English Dominican Province. Vol. 1. New York: Benziger, 1911. (268-270)
Bisping, August. Exegetisches Handbuch zu den Briefen des Apostels Paulus: Erklärung des Zweiten Briefes an die Thessalonicher, der Drei Pastoralbriefe und des Briefes an Philemon. Vol. 3, Issue 1. Munster: In der Aschendorff’schen Buchhandlung, 1858.
John of Damascus (Ἰωάννης ∆αµασκοῦ). "An Accurate Exposition of the Orthodox Faith" (Ἔκδοσις Ἀκριβὴς Τῆς Ὀρθοδόξου Πίστεως). Book 2, Ch. 29. Patrologiæ Cursus Completus: Series Græca Prior. Ed. Migne, Jacques Paul. Vol. 94. Petit-Montrouge: Imprimerie Catholique, 1864. (963-970)
Thayer, Joseph Henry. A Greek-English Lexicon of the New Testament: Being Grimm Wilke’s Clavis Novi Testamenti. Rev. ed. New York: American Book, 1889.