Well is 'eu' in Greek, born is 'genios' or 'genia' in the feminine. Eugene/Eugenia mean 'well-born' and are completely unconnected to Hermione.
Hermione seems clearly formed on Hermes (not just based on the letters but on other existing Greek words, it's a consensus opinion of Greek scholars) but the exact formation isn't clear. The most likely root of Hermes is Herme/Herma which means 'support'. Hermax meant 'cairn, heap of stones' and herma meant 'prop, support'. Hermes was also the god of roads and roadside shrines to him, hermeia, were originally just heaps of stones that travellers would add to. These were elaborated and then Christianised - on mountain paths of Greece today you'll find roadside shrines to saints, the descendants of the old hermeia.
Hermeion was 'shrine of Hermes', -eos was a word meaning 'his, hers, his or her own'. So perhaps the closest we can get to a meaning for Hermione is to put them together Hermeione for 'Hermes' own shrine' or getting very liberal, going off the root meanings only for 'her own support'. Actually, 'her own support' would be very like the Hermione's I know - all a strikingly independent bunch.
If a book doesn't tell you *how* they came to their conclusion, it's usually because they *can't*, having dreampt the whole thing up out of thin air or very flimsy evidence.