Elizabeth and Catherine are both wonderful classic names with lots of scope for nicknames. Isobel and Sophie are also lovely.
Lizzie isn't dated and both the Lizzie's I know have been absolutely kick ass and seriously impressive so I don't have any negative associations.
Bear in mind, regardless of which nick name you choose ultimately it's the child who will choose their nickname.
Eg my DS loves his full name but only allows us to use the short form, everyone else is very politely told to use the full form and as a result will tend to call him the full form mostly too.
So you may choose Lizzie or Katie and she'll turn round one day she'd 7 or 8 and say from now on she's going to be Lisbeth or Kit. .