There is a very detailed examination of this issue by Margaret Mowczko from an egalitarian perspective (answering in the affirmative) posted here:
Was Phoebe a Deacon of the Church in Cenchrea? (Part 1)
Was Phoebe a Deacon of the Church in Cenchrea? (Part 2)
While the usage of the word διάκονος in scripture is discussed at some length, the strongest arguments in favor are the extra-biblical witnesses that there were women known as deaconesses/ministers in the early Church of which Phoebe was a noteable example:
Pliny the younger, who was governor of Pontus and Bithynia in 111-113, wrote to Trajan saying that he had tortured two female slaves “who were being called ministers” (quae ministrae dicebantur). (Letters 10.96.8) Elizabeth McCabe (2009) notes that the Latin word ministra is synonymous with the Latin word diāconus, and that a diāconus can be defined as a minister of the church, that is, a deacon. Pliny believed that the two tortured women were official ministers, or deacons, of the church.
Origen lived in the years 185-253, a time when ordained female deacons were active in the church. (Campbell 2009:61) In around 246 Origen wrote his commentary on Romans (the oldest commentary on Romans that still survives) and it is apparent that he believed Phoebe to have been an official female deacon. In reference to Romans 16:1-2 he declared that “This passage teaches by apostolic authority that women also were appointed (constitiu) in the ministry of the church (in ministerio ecclesiae), in which office Phoebe was placed at the church at Cenchrea . . . And therefore this passage teaches two things equally and is to be interpreted . . . to mean that women are to be considered ministers (haberi . . . feminas minstras) in the church.
As well as literary evidence, there is epigraphic evidence which indicates that Phoebe was famous and regarded as an ordained deacon by the Early Church. A funerary stele from the Mount of Olives in Jerusalem, and dated to the latter half of the 4th century, or possibly later, reads: “Here lies the slave and bride of Christ, Sophia the deacon (hē diakonos), the second Phoebe (hē deutera Phoibē), who fell asleep in peace . . .” (Horsely) To be referred to as “the second Phoebe” was clearly meant as an honour for the deacon Sophia being commemorated on the stele.
A somewhat lengthy but highly pertinent quote addressing 1 Timothy 3*:
Diakonoi are mentioned in First Timothy, one of the later New Testament letters. The diakonoi of 1 Timothy 3:8ff were probably official deacons. What is less clear is who the women mentioned in 1 Timothy 3:11 were. Whether these women are female deacons or the wives of deacons is debated, but, considering that up until the fourth century there was no separate word for female deacons, it could be that the female deacons were simply called “women” here to distinguish them from the male deacons.
There are indications in the text which suggest that these women were female deacons and not deacon’s wives. For instance, there is no mention of the wives of overseers (or bishops); and it doesn’t make sense that the writer of 1 Timothy would regard the moral requirements of deacons’ wives to be worthy of mention, but not those of overseers’ wives. Also, if deacons’ wives were intended, we would expect a definite article or a genitive pronoun in the Greek of 1 Timothy 3: 11 (which could be translated as “the wives” or “their wives” respectively.) However, it is the use of the word “likewise” (hōsautōs) which indicates that a distinct but similar group is being addressed in verse 11. (Giles 1989:61)
“Likewise” (hōsautōs) is found at the beginning of 1 Timothy 3:8 and 1 Timothy 3:11. Massey (1989:61) writes that “likewise” is “customarily used to introduce the second and third entities in a series.” He suggests that the use of hōsautōs “seems to place the three groups [overseers, male deacons, and women] in categories of a similar nature.” That is, the people belonging to these three groups are involved in somewhat similar ministries and require similar moral qualifications. Taking the word “likewise” into account we can see that verses 8-10 refer to the male deacons, verse 11 specifically refers to the female deacons, and verses 12-13 probably refer to both the male and female deacons.
John Chrysostom weighed in on the debate about whether the women in 1 Timothy 3:11 were deacons or not. In his Homily 11 on 1 Timothy he wrote: “Some have thought that this is said of women generally, but it is not so, for why should he introduce anything about women to interfere with his subject? He is speaking of those who hold the rank of deaconesses.” In response to 1 Timothy 3:12 (including the idiomatic phrase “a one woman man”) he added “This must be understood therefore to relate to Deaconesses. For that order is necessary and useful and honourable in the Church . . .”
*This is a passage where a masculinist bias in a majority of English translations is particularly evident - I personally find it fascinating and illuminating to read a translation that preserves the gender neutrality of the original Greek.