Olive and Olivia

Olive
Meaning is obvious - olive, from Latin oliva. Since Greek times the olive branch has been the symbol of peace, so it could also be taken to mean peace - though it does not directly.
In terms of connection there is Olive Oyl (and olive oil) - the girlfriend of Popeye - probably what has condemned Olive and allowed Olivia to flourish.

The olive plant (as shown to the right) is grown in Eastern Mediterranean. The fruit of the plant is initially inedible and must be treated before eaten. This involves being soaked in a solution of NaOH and thoroughly washed with water to remove oleuropein - a naturally bitter carbohydrate. Green olives are unripe olives that have been processed and allowed to ferment. Black olives are ripe and unfermented thus tasting milder. Olives have also been used since ancient times for making Olive oil.

Olivia
This is a Latinate name, first used by Shakespeare for the rich female heiress in Twelfth Night - wooed by Orsino, falls in love with Cesario (aka Viola as a man) and finally marries Sebastian, twin of Viola. This romantic connection (much more romantic than being the girlfriend of a spinach-muncher), the fact that it's a female form of Oliver and shares the same nicknames (Ollie, Livvy) means that it's popular in mainly English speaking countries - UK, USA, Australia, NZ and (interestingly, seeing as I don't know much about their naming trends) Finland.