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My first inclination would be to try: teknia NEAR paidion

That found this paragraph in Wuest's Word Studies in the Greek New Testament:

“Little children” is paidion (παιδιον). Vincent says: “Compare teknia (τεκνια) little children (v. 1), which emphasizes the idea of kinship, while this word emphasizes the idea of subordination and consequent discipline. Hence, it is the more appropriate word when spoken from the standpoint of authority, than of affection.” Teknia (Τεκνια) is related to tiktō (τικτω), “to give birth to,” and emphasizes the birth or genital relationship, whereas paidion (παιδιον) is related to paideuō (παιδευω), “to train children.” Our word “pedagogue” comes from the latter word. One could translate paidion (παιδιον), “little child under instruction.”

The "Vincent" he is referring to is Marvin R. Vincent, whose Word Studies in the New Testament is also available in Logos, but doesn't say anything more on the matter than that quoted bit.

Next I'd use the Greek equivalents to search:τεκνια NEAR παιδιον

This finds the same hit and nothing else. So I'd try expanding the distance between the words:τεκνια WITHIN 30 words παιδιον

This adds another significant hit in Word Biblical Commentary Vol. 51 (1, 2, and 3 John) by Stephen Smalley:

"γράφω ὑμῖν, τεκνία. One immediate problem posed by this phrase, and the parallel phrases in vv 13–14, is the precise meaning of the address which John uses for his readers: here, “dear children” (v 13a, “fathers”; v 13b, “young men”). The Gr. words used are τεκνία (“dear children”), πατέρες (“fathers”) and νεανίσκοι (“young men”); although during the virtual repetition of vv 12 and 13 in v 14 παιδία (“little children”) is used instead of τεκνία. (For τεκνίον, “child,” see 2:28; 3:7 in some MSS, 18; 4:4; 5:21; for παιδίον, “child,” see 2:18.) Against Westcott (61), who thinks that τεκνίον emphasizes “the idea of subordination,” while παιδίον suggests “kinsmanship,” there is probably no signficant difference involved between the two words. Both are terms of endearment. Brooke, 43, argues that τεκνία, as used here, may express “community of nature,” in contrast to παιδία, which implies (as in 1 Cor 14:20) the need for moral training and guidance. But the παιδία who have “known the Father” (v 14), or who are told that the last hour is upon them (2:18), and the disciples (παιδία) of whom the risen Jesus inquires, “Haven’t you any fish?” (John 21:5) can scarcely be considered as morally deficient! The diminutive τεκνία is confined in the NT to the Johannine literature (the reading of Gal 4:19 is uncertain), and is used only once in John’s Gospel (at 13:33)."

The "Westcott" that he's referring to is Brooke Foss Westcott whose The Epistles of St. John: The Greek Text is available for preview on Amazon.com. Here's the full excerpt Smalley is disputing:

"παιδία] infantes V., pueri Aug., little ones. This title, little ones, which like τεκνια is applied to the whole Christian body, differs from little children by emphasizing the idea of subordination and not that of kinsmanship. St John speaks not as sharing the nature of those to whom he writes, but as placed in a position of authority over them. Comp. v. 18 (John xxi. 5).
"In correspondence with this difference in the address St John gives a different reason for his writing: because ye know the Father.
"The sense of an  immediate personal relationship to God (comp.  John xiv. 7) gives stability to all the gradations of human authority. In this respect 'knowing the Father' is different from 'knowing Him that is from the beginning.’ The former involves a direct spiritual connexion: the latter involves besides an intellectual apprehension of the divine 'plan.' The knowledge 'of the Father' is that of present love and submission: the knowledge of Him 'that is from the beginning' is sympathy with the Divine Thought which is fulfilled in all time."