shinee - the english word shine + the korean suffix ee, making it mean "the one who receives the shine/light"
twice - they charm you twice: once through the ears and then through the eyes. also, within the context of the fandom name "once", it means that they repay twice the love their fans give them
fromis_9 - from is, is standing for idol school, the survival show they debuted through. the 9 represents the number of members they had at debut.
nine.i - at debut, each member represented one kind of energy??? i think??? except for vari, the connector (represented by the i), whose role is to connect the nine energies. now that they're down to nine members in total, including vari, the lore doesn't really stand anymore