It's possible this name was first introduced by fans of the Japanese animated films known as "anime". The first reference to a character named Journey on the Internet Movie Data Base is to an anime film called "Gall Force: Destruction" which came out in 1987:
Perhaps the original Japanese name of the character either sounded like "journey" or had a similar meaning in the original Japanese. But we would have first to find out what the original Japanese name was and then get someone who knew Japanese characters to interpret it.
Journey is also the name of the title character in a "young adult" novel by Patricia MacLachlan, published in 1991:
Once the name was "legitimated" by being a character name, it may have increased simply because of the "different but not too different" factor, as its sound is close enough to names like Jenny and Courtney that some parents might have been attracted to it.
I realize that the film character is a boy and Journey is more popular for girls, but that may have been an inevitable shift given the name's ending in -ney. The anime character does seem to be female.