I like Catherine as much as I like Elizabeth, Cat/Cate are good nicknames but Cathy/Cath have the same "dated" feel as Lizzie or Liz. IMO anyway!
Isobel and Sophie are more popular at the moment but I don't think they will ever not be commonly used, they're classics.
Please be aware if you introduced your daughter to me as "Lizzy" I wouldn't even think "oh gosh that's dated" let alone say it, I just wouldn't use the name for a child of my own. If you like it go for it!