I'm considering upgrading to v. 7.x and wanted to know if there is an option for individual members to customize and/or enable/disable the sidebar? I looked through my "My Stuff" section briefly and didn't see anything that would allow this.

I assume you are talking about islands. That is what the features are called such as the items on the right here.(Search,Who's online,Shoutbox)
If you enable the pertinent members settings in the control panel a member can elect to hide the left side or the right side or both sides.
But a user can't decide to exclude just the shout box as an example for this site.

This user ability appears to be disabled at this site from the control panel.

Okay, they used to be called "Pal boxes" back when.

Thanks for responding and confirming that an individual member has the option to not display the sidebar(s). I didn't realize that there were two, left and right, so I learned a bit more than I asked for.