If your favorite mascara has dried up, chances are it's old anyway. I would go and re-purchase a new one. Mascara really can get some nasty bacteria and microscopic monsters crawling around on it even when it only has a little age on it. So if I were you, I'd pitch it.
Try a drop or two of water. Not much. Then pump the applicator a bit. You should be able to keep the mascara lasting long enough to get some more. But did you know you should replace your mascara every 3 months to prevent eye infections and when it becomes difficult to get the mascara out or it dries up, it's time to replace!