The messages indicator only appears when an application has registered to use it; by default, no applications are registered to use it, meaning that the messages indicator is hidden. However, some applications, such as Thunderbird, register with the indicator as soon as you open them and leave no apparent way to remove them. (It's possible there's an option buried somewhere in Thunderbird's Preferences menu, but I haven't looked too thoroughly.)
My solution is to reset the list of applications registered with the indicator to the default (empty), then kill the indicator to refresh it. (For some reason, the messages indicator doesn't seem to automatically refresh itself, unlike all the other indicators, and pretty much everything else in dconf for that matter...)
Open a terminal and copy/paste the following:
dconf reset /com/canonical/indicator/messages/applications
killall indicator-messages-service