I'm using LL 2.6 LTS and have found a way to make this work using some of what has already been documented, however, I don't want Dropbox running all of the time.  I only want to load it when I want to upload items to share with friends and relatives.  After the upload completes, I typically shut down Dropbox.  
Using terminal as others have described is a bit cumbersome, so I decided to modify the original item in the Menu instead. I followed the steps below:
From the Desktop, Right click on the 
Menu button on the left end of the taskbar
Left click on 
Edit Applications to open the Main Menu window
Left click on 
Internet in the left column of the Main Menu
Look for 
Dropbox on the right column and left click it to highlight it
Left click on the 
Properties button on the right edge of the Main Menu to open the Launcher Properties
Keep 
Name: Dropbox
Replace 
Command: bash -c 'export DBUS_SESSION_BUS_ADDRESS="" && dropbox start -i' 
Keep 
Comment: Sync your files across computers and to the web
Make sure the box is UnChecked next to 
Launch in Terminal?
Click the 
OK button
That's it !
So far, it works like a dream and the icon comes up in the task bar like it should and I can use the menu and quit Dropbox the way I always have by clicking on the Dropbox Icon in the task bar by the clock and then selecting Quit Dropbox when I'm done using it.
Hope this helps !
 
