The nearest you can get to this is to use a browser action/button and use the tooltip (a.k.a. title) to display the information when the mouse hovers over the button. For example, in your background.js file your code would look something like:
appAPI.ready(function($) {
appAPI.browserAction.setResourceIcon('icons/icon.jpg');
appAPI.browserAction.setTitle('Information displayed on mouse hover');
appAPI.browserAction.onClick(function() {
alert('Hello world!');
});
});
For more information, see appAPI.browserAction.
[Disclosure: I am a Crossrider employee]