I suppose that this would mean one of my typical setting takes ("Bahamut is the father of all evil dragons, Tiamat is the mother of all good dragons") applies less.
Probably the most interesting way to proceed forward would be "so, what would happen if you unified their themes as part of a united pantheon?"
It would probably mean that the Drow would be more open to consorting with dragons. it would probably lead to awesome web-spinning spider-dragons. "Giant webs" allow me to make pretty cool elaborate fight scenes, so I'd break them out a heck of a lot.
I don't know if there are any subterranean dragons yet, but if there aren't, I would make white dragons subterranean dragons*, and have Drow ride them into battle! They'd still have cold breath, since the earth is associated with cold for me.
So yeah, you could do something interesting with it. Wouldn't be my gut instinct- I like having Tiamat being not unheard of for worship in the civilized world, especially among adventurers. Lolth is kinda crazy.
(*:My drow are albino, for some reason. My surface elves are dark-skinned.)
Edit: Now, Tiamat and Erathis being lovers, that's some ham-fisted thematic resonance right there!