My two coppers:
It's possible that someone created a Priestess of Lolth class. I think I've seen several speciality priests (like Doomguide of Kelemvor) as base classes, not too long after 3e was released. But you actually don't need such a thing
And Eremite's right. With that background, a cleric isn't really fitting. That girl didn't have any formal cleric training (Lolth's priestesses spend several decades in the academy, at least in Menzoberranzan), no indoctrination and all. She's no cleric. Favoured soul seems to fit like a glove: You didn't choose Lolth, Lolth choose you. You're a Drow princess, but the humans took you away. Now she wants you back. So you get some nice promotional gifts from her, in the hope you subscribe.
Beside that, you might play something else (say, rogue, fighter), and later become a divine champion or divine seeker (or something like that), when the call becomes stronger. Maybe you can get your DM to write a special feat
Something like "Lolth's Call (can only be chosen at 1st level). The spider queen calls you. Once per day, you can cast talk with spiders, which works like talk with animals, only with spiders. You can also cast web once per day.
Or something like that.