Yes, you can! All you have to do is use any image you want, for example:
<img src="custom-facebook-image.png" />
And then associate it with the Facebook URL as follows:
http://www.facebook.com/sharer.php?u=http://www.example.com&t=CustomTitle
Just make sure you escape all the characters in the url and you can add it as href:
<a href="http://www.facebook.com/sharer.php?u=http://www.example.com&t=CustomTitle">
<img src="custom-facebook-image.png" />
</a>
source
share