The passage of Scripture you quoted has been often used as a "prosperity" passage; when a "soul" prospers(his spiritual communion with God), his health prospers and the conditions of his life prosper(relationships, family, finances, etc.)
But since the question now asks for the hermeneutic/ process I quote the following source:
*Hugiaino The verb hugiaino in classical Greek means “to be physically
or mentally healthy or sound.” This idea of soundness extended to
other areas as well where it was used of soundness in political or
religious matters. It was used in the closing of letters and meant
“goodbye.” The word occurs 24 times in the Septuagint where it
virtually occurs in some form of greeting. It appears 12 times in the
New Testament. A Greek-English Lexicon of the New Testament and Other
Early Christian Literature-Third Edition: (1) to be in good physical
health, be healthy, lit. Mt 8:13 v.l.; Lk 5:31 (Artem. 4, 22 ! '
$ιαυι ( ' #μυι "' αι)); 7:10; 15:27. As a
formula in an epistolary greeting (e.g. Ltzm., Griech. Papyri2 [=Kl.
T. 14] 1910 no. 1, 3 [=BGU 423]; 2, 3 [=BGU 846]; 8, 3 [=BGU 27]; 9, 4
[=BGU 38] and oft. in pap; cp. EpArist 41) 3J 2. (2) to be sound or
free from error, be correct, fig. in the Pastoral Epistles w. ref. to
Christian teaching: $ιαυα "ι"α#αα 1 Ti 1:10; 2 Ti 4:3; Tit
1:9; 2:1. $ια ι 1 Ti 6:3; 2 Ti 1:13. $ιαι () *
ι Tit 1:13; 2:2 (on its use w. the dat. cp. Jos., C. Ap. 1, 222).
Cp. $ι Tit 2:8 ($ι 2). Thus, in accord w. prevailing
usage, Christian teaching is designated as correct instruction, since
it is reasonable and appeals to sound intelligence (Plut., Mor. 2f
. (Page 1023) Louw and Nida list two meanings: (1)
the state of being healthy, well (in contrast with sickness)—‘to be
well, to be healthy.’ $ιαω (23.129) (2) to be correct in one’s
views, with the implication of such a state being positively
valued—‘to be correct, to be sound, to be accurate.’ (72.15)
(Greek-English lexicon of the New Testament: Based on Semantic
Domains; Electronic ed. of the second edition; 39.1; 1996; New York:
United Bible Societies*(taken from here)
The author further states:
In 3 John 2, the verb hugiaino means “to be health” or “to be of sound
health” referring to sound physical health. The present tense of the
verb is a customary or stative present referring to an ongoing state.
Thus this tense refers to the state or condition of being in good
health. The active voice of the verb is stative expressing the same
thing as the present tense. The infinitive form of this verb hugiaino
is an infinitive of purpose meaning that it is indicating the purpose
of the action of its controlling verb. It is answering the question as
to “why” John prayed for Gaius(1)
Given the usage of this word "hugiano" and it's common usage in NT greetings, it can be inferred that it is typical form of greeting that John wishes on his disciples, and not a particular inquiry as to the health of the specific disciple, Gaius.
Since the Early Church Fathers determined that these letters were indeed canon, having passed the tests of authenticity and doctrinal purity,(2)
they are as Paul states in 1 Tim. 3:16,
All scripture is given by inspiration of God, and is profitable for
doctrine, for reproof, for correction, for instruction in
It is therefore right to apply the timeless truths of Scripture, regardless of who they are addressed to. These are as 2 Pet. 1:3-4,
According as his divine power hath given unto us all things that
pertain unto life and godliness, through the knowledge of him that
hath called us to glory and virtue:
4 Whereby are given unto us exceeding great and precious promises:
that by these ye might be partakers of the divine nature, having
escaped the corruption that is in the world through lust.
Therefore, they are meant for today as well as the time they were written.
- More information can be found on the word usage at the source mentioned
- Discussion of authenticity of 3 John can be found here.