Τεκνία is the plural diminutive of τέκνον (child).
diminutive adjective (14th century) 1 : indicating small size and sometimes the state or quality of being familiarly known, lovable, pitiable, or contemptible — used of affixes ... -- Merriam-Webster, I. (1996). Merriam-Webster’s collegiate dictionary (10th ed.). Springfield, MA: Merriam-Webster.
The general consensus is John isn't using the term to express little by to express loving familiarity.
τεκνίον, ου, τό (Epict. 3, 22, 78; Pal. Anth.; PFlor. 365, 15 [III AD]; POxy. 1766, 14) dim. of τέκνον; (little) child, voc. pl. τεκνία; in our lit. only in the voc. pl., used by Jesus in familiar, loving address to his disciples, or by a Christian apostle or teacher to his spiritual children τεκνία J 13:33; 1 J 2:12, 28; 3:7, 18; 4:4; 5:21. τεκνία μου (Test. Reub. 1:3 v.l.) Gal 4:19 v.l.; 1J 2:1. M-M.* -- Arndt, W., Gingrich, F. W., Danker, F. W., & Bauer, W. (1979). A Greek-English lexicon of the New Testament and other early Christian literature : a translation and adaption of the fourth revised and augmented edition of Walter Bauer’s Griechisch-deutsches Worterbuch zu den Schrift en des Neuen Testaments und der ubrigen urchristlichen Literatur (p. 808). Chicago: University of Chicago Press.
What is difficult with this type of translation is it is often geographically dependent. What are better terms ro use? Maybe something like kiddos.
A problem is English doesn't seem to have a vocative endearment term with the meaning children. Spanish has "Hijitos míos," a very good fit. To further illustrate this, if you enter children in Google translate, you get niños, but if you enter my children, you get Mis hijos.