The general sense of the aorist aspect is that is provides an external (rather than internal or perfective-stative) portrayal of the action.
[The action is] viewed as a whole from the outside, without regard for the internal make-up of the occurrence.1
That is not to say you won’t find commentaries reading something of the internal make-up of the action into the aorist. In 1972, Frank Stagg was able to say,
To the grammarian it may seem like beating a dead horse to protest that the aorist does not necessarily reflect the nature of the action or event it covers. But the horse is not dead; he is very much alive and cavorting rather freely in the exegetical and theological pastures. The fallacy of the "theology of the aorist tense" stubbornly persists, even in the writings of distinguished scholars. 2
You’ll still see this sort of argument around. Generally the representation of the aorist is along the lines of “punctiliar” (momentary) or “once-and-for-all”. As Stagg points out, the actions thus portrayed are indeed punctiliar, but it is not the aorist aspect as such that makes them so. Applied to John 3:16, the aorist inflection of ἠγάπησεν does not indicate that he loved the world once or only once (nor does it indicate a past, present, or future orientation of the action). It simply provides the action, viewed from the outside, inflected for subject.
Of course, the verb and its aspect due not exist in a vacuum, and there is something to be said about the nature of the action based on the linguistic and narrative context. The most important part of the local context is the οὕτως....ὥστε construction in which it is situated. That may be roughly translated, “In this way [God loved]....such that [he gave]”. That construction emphasizes the effective nature of the love and “insists that the envisaged consequence really did ensue”.3,4
Is it referring specifically to the giving his special son?
Yes. The verb is contained within the οὕτως clause. The emphasis is on the effect of the action depicted in the ὥστε clause (
ἔδωκεν...| he gave...) that follows. This is contained within a larger cause-effect structure (
...ἵνα... | ...in order that...) describing the purpose of this action (
πᾶς ὁ πιστεύων...ἔχῃ ζωὴν αἰώνιον | all those believing...might have eternal life).
For more on verbal aspect in John 3:16, see In John 3:16, is the word believe a continous action for everlasting life? For more on the general topic of the aorist verb (note: off-topic here), see What Is The Aorist Tense Of A Verb?
1. Buist Fanning. Verbal Aspect in New Testament Greek, Oxford: Clarendon, 1990, 97.
2. Frank Stagg. The Abused Aorist Journal of Biblical Literature. Vol. 91, No. 2 (Jun., 1972), 222.
3. D.A. Carson, The Gospel according to John, (Pillar NT Commentary) Eerdmans, 1990, pp. 204ff.
4. English translations, following the KJV, often use “For God so loved....” which is interpreted by most modern (at least American) English speakers as “God loved so much...” Thi is not the type of “so” contained in οὕτως (rather, “in this way”). However, it turns out to not be far off given the larger structure of the sentence.