Laudator Temporis Acti

Saint Augustine, Confessions 9.12.32 (after the death of his mother, tr. J.G. Pilkington):

It appeared to me also a good thing to go and bathe, I having heard that the bath [balneum] took its name from the Greek βαλανεῖον, because it drives trouble from the mind. Lo, this also I confess unto Thy mercy, "Father of the fatherless," that I bathed, and felt the same as before I had done so. For the bitterness of my grief exuded not from my heart. Then I slept, and on awaking found my grief not a little mitigated...

visum etiam mihi est ut irem lavatum, quod audieram inde balneis nomen inditum quia Graeci balanion dixerint, quod anxietatem pellat ex animo. ecce et hoc confiteor misericordiae tuae, pater orphanorum, quoniam lavi et talis eram qualis priusquam lavissem, neque enim exudavit de corde meo maeroris amaritudo. deinde dormivi et evigilavi, et non parva ex parte mitigatum inveni dolorem meum...

James O'Donnell, in his commentary on the Confessions, doesn't elucidate this passage, beyond correctly noting that the derivation is spurious.

Isidore, Etymologies 15.2.40 (tr. Stephen A. Barney) repeats the spurious derivation:

Baths (balneum) are assigned their name from the idea of the lifting of sorrow, because the Greeks called it βαλανεῖον (cf. βάλλειν, "cast away"; ἀνία, "grief"), since it takes away one's anxiety of spirit.

balneis vero nomen inditum a levatione maeroris; nam Graeci BALANEION dixerunt, quod anxietatem animi tollat.

Hjalmar Frisk in his Griechisches Etymologisches Wörterbuch says that there is no satisfactory etymology of Greek βαλανεῖον ("unerklärt"). Apparently Pierre Chantraine in his Dictionnaire etymologique de la langue grecque (unavailable to me) suggests that it comes from βάλανος (acorn) because bath stoppers were shaped like acorns, but Robert Beekes in his online Greek etymological dictionary calls that an "improbable" suggestion.

René Ginouves, Balaneutike: Recherches sur le bain dans l'antiquité grecque (Paris: E. de Boccard, 1962) has no index, and the table of contents doesn't lead me to think that he discusses the etymology.

I wonder where Saint Augustine got the idea. He admits (Confessions 1.13.20 and 1.14.23) that he didn't learn much Greek as a boy. Pierre Courcelle, Recherches sur les Confessions de Saint Augustin, new ed. (Paris: E. de Boccard, 1968) is not available to me. Varro knows that Latin balneum comes from Greek βαλανεῖον, but he doesn't discuss the derivation of the Greek word.

Thanks to Fr. Gerard Deighan, whose email on this subject prompted me to look further into it.