I like Oliva Rose the best. The nicknames i have heard for Olivia are Liv, Livvy, are probably the most common.
The problem is that when you think of nicknames they may not alwasy get used and someone will come out with something quite differnt when they are born. My DD is Brearna Lee (Brea - R - Na) and i always thought her nickname would be Brea, Bee or Bumble Bee but no she has had NONE of these and she gets called Banana, Narna or Bana totally nicknames that i never thought i would here...not that i mind as they are pretty cute. But i am sure once she gets older she will get Brea and who know what else when she gets to school.