I tested a bit of this a few years ago, using a primarily German save editor call FF7SaveEdit.
It can now be done using black Chocobo. Basically, edit your save file so that in the PHS the character is "Available to switch". This was, you can have a Cloudless party. Sometimes the character that appears onscreen is random, but there is always a reason.
For example, only Cloud, Tifa and Cid have programming for the world map. If you're on the world map and you create a party that has none of them in it, when you exit the menu the character that appears is the last character you used on the world map. So if you say put Tifa, Red XIII and Yuffie in the party, then swapped Tifa for Vincent, then Tifa would appear on the world map.
Conversely, if there is a party in a town (such as Cosmo Canyon) without Cloud, you can control almost any character. The character that appears is usually the LAST character you swap in in the PHS. I had a thread about this a while back.
Sometimes if you enter a town where no paths are written for any of the characters in your team, the game will hang, or you'll be invisible and get stuck. Any character that is not Cid, Tifa or Cloud will encounter problems at some point, because paths aren't written for them in various locations. To combat this, me and (I think it was Ultima Espio) were talking on MSN about adding paths for all characters using Meteor, which he tried. Needless to say, it was really buggy. I think he ended up with Biggs or Jesse appearing in the forgotten city. It was a long time ago.
I'll find the thread and edit this post to include it. There's more information in there.
EDIT:
Here is the original thread from 2008
