Changing the web browser invoked by Thunderbird
From MozillaZine Knowledge Base
When you click on a web link in Thunderbird, it invokes the default web browser for the operating system. You could change the default browser for your operating system but that effects other applications. Usually the best solution is to specify what browser Thunderbird can use with the network.protocol-handler.app.http and network.protocol-handler.app.https settings. While that setting is normally associated with Linux, it will work under Windows and OS X too.
Use Tools -> Options -> Advanced -> General -> Config editor to launch the config editor, right click on its listing, select New, select String, enter network.protocol-handler.app.http and then the complete path to whatever browser you want to use. If the setting already exists just double click on the setting to change the path. Do the same thing for network.protocol-handler.app.https to support secure connections. Some browsers support optional command line arguments such as specifying private browsing mode and opening the web page in a new tab. For example: "C:\Program Files (x86)\Mozilla Firefox\firefox.exe" -private -new-tab
Recent versions of Thunderbird may also require you to modify network.protocol-handler.warn-external.http and network.protocol-handler.warn-external.https to true. Those are Boolean settings. If you need to do that it will popup a window asking you to select what browser to use when you click on a link in a message. Select the browser and check the checkbox to have it remember the setting. If you change your mind and want to use the default system browser right click on the settings you modified in the config editor and chose reset from the context menu. Check in Tools -> Options ->Attachments -> Incoming to see if it added a http and a https entry. If it did, select "Delete Action" as the Action and press OK. Then restart Thunderbird. 
The Open With add-on lets you specify what browser to use to open a link. However, you have to select that browser each time as the add-on is designed to support testing, not somebody who just wants to use a different browser than the default web browser with Thunderbird.