The thing is with a name like Elizabeth you won't get to choose the nickname, well you might when they are tiny but nicknames tend to evolve so if you don't like ALL the nicknames shortening then I would be wary of choosing it.
We were going to call our dd Genevieve, we would have shortened it to 'vivi' but I knew it would become Jen/gen which I don't like at all. Even when we mentioned it as a possible name once baby was born people shortened it straight away. Dd was then nameless for two weeks when she became Merryn :)