Eilistraee has always been Lolth's daughter, since Ed Greenwood created her for his own Forgotten Realms, and since when TSR asked him to make her official (in 2e). To my understanding, she has always been a dark elven goddess of beauty, song, dance, hunt, and swordwork (the bastard sword has always been her favorite), even tho she only picked up moonlight far later, after the drow were cursed, according to Ed's idea of her (she did so because the night was the only time when new converts could move on the surface safely, and so her preferred time to appear to them).
Lolth is older than Eilistraee, however Eilistraee and Vhaeraun (twins) became known to the Torilian dark elves before Lolth, because Vhaeraun was exiled there, and Eilistraee chose to wander among the drow there after her mother's banishment. Lolth was instead sent to the Abyss, but a fortuitous event brought her attention to Toril. Thanks in no small part to the elves weakening Vhaeraun's influence in Ilythiir through the First Sundering, and--far later--thanks to the elves' extermination of most of Eilistraee's followers in Miyearitar (the Dark Disaster), which reduced her influence to nearly a non-factor, Lolth's plan successfully made her the most influential drow deity when the dark elves were cursed and exiled.