Most often when Bible passages are discussed in relation to the office of deaconess, the starting point for discussion is Romans 16:1. Wilhelm Loehe - in a series of paragraphs titled "On the Deaconess" which he wrote in 1858 - was additionally very interested in the application of 1 Timothy 5:3-16 to the female diaconate.
Loehe wrote, "Holy Scripture contains a passage, which does not, to be sure, talk about the office of the deaconess, but has nonetheless been interpreted as talking about the deaconess office from time immemorial; it is the passage 1 Timothy 5:3-16. When we have just said that the passage has been interpreted as talking about the deaconess office from time immemorial, we must not think of a unanimous and general interpretation of the Christian Church. Two great teachers of the ancient church who were in office during the second half of the 4th Century, the oriental John Chrysostom, patriarch of Constantinople, and the occidental Ambrose, archbishop of Milan, see the adduced passage talking about nothing else but the care of widows. Contrariwise, others, like the occidental Tertullian, who died in 220, and the oriental Epiphanius, who died in 403, already hold that the passage talks about the vocation of deaconess. The view prevailed in later times and has become the generally accepted one. If the passage talked only about the care of widows, one has said, it would be hard to see, why the selection of the widows to be cared for was so tightly attached to certain moral and other virtues, since Christian mercy does not make such great distinctions; therefore widows have to be in mind who, on the one hand, to be sure, are cared for, but who, on the other hand, are employed for the blessing of the congregations. There is something to commend this view, and the entire antiquity has therefore taken the requirements for the deaconess office from this passage."
[Quotation taken from Wilhelm Loehe on the Deaconess, translated by Holger Sonntag.]