What would be a good Latin noun or verb to translate "chat" in the sense that would apply to our chat room, for example? There are verbs like garrire, fabulari, verbigerare, and simply colloqui but I don't know which one would have the most suitable tone.
Your suggestion garrire is perfectly all right for inconsequential chatter. Colloqui is pretty generally applicable, but there's really no shortage of options to use for 'chat'. A few are:
res cum aliquo agitare; de rebus cum aliquo colloqui; sermonem cum aliquo conferre.
I particularly like [inter sese] vicos et urbem laudare, which I remember from my schooldays and which think is from Horace, Ser. I, 9.