I would recommend buying one of the pens that they sell at WDW for $3. I know it's a crazy amount of money to spend on a pen, but if you are asking what the characters prefer, those pens are the best. (Trust me, I know)
The problem that characters have with the Sharpies (and other pens of this type) is that more often than not, they are permanent markers. The character costumes cost thousands of dollars each and when you hand the character a permanent marker, they are forced to focus more on the pen, than on the child. Imagine that you are wearing a brand new outfit that costs hundreds of dollars and someone hands you a full cup of grape juice. You will naturally be very attentive to the cup of grape juice as to not get anything on your new outfit. The same principle applies to the characters. They will try their best to keep their focus on the show, but they will need to spend extra attention to the props instead of the guest. This hurts the show and ultimately may cause the experience to lack magic.
Also, I've noticed that some of the markers people use will bleed through the paper onto the next page or will leak into the cap so when the marker is handed to the character, they MUST grab the pen and get ink on their fingers.
Definitely have the pen ready and the book open to the page you want them to sign. Also be sure to hand the book to the character so that it is not up-side-down. It will help them not to have to adjust the book and take time away from interacting with the guest.
Just a few thoughts.