To rename computers, Dropbox users shouldn't have to go through the trouble (and associated risk) of unlinking and relinking their Dropbox accounts. This is the process recommended by Dropbox Support here in the forums.
For some, this is a horrible workaround, especially on a computer with a large number of files which gets shuffled around by Dropbox when disabling Dropbox backups. In addition, there's the long amount of time it takes to Dropbox to relink and "process" the files.
And, this didn't even work for me! Unlinking Dropbox, signing out, rebooting and then relinking AFTER I have renamed my Mac did not work. Dropbox automatically went with the previous name "Mac (3)" which is not helpful at all. Where is Dropbox getting the old name from if I've renamed my Mac in Sharing Prefs, unlinked and rebooted???
I love Dropbox for the most part, but this workaround is not an acceptable solution. 
The workaround didn't work, and now I have to wait as Dropbox "processes" hundreds of thousands of files. All for nothing.