First, mice don't need meat to eat.
Second, deli meats are super high in salt, and thus not a healthy thing to be feeding rodents.
I used to give my rodents the leftover chicken "bones" with the bits of meat on them after *I* ate chicken. Most "leftover" meats you could probably give them, but again... mice don't need meat and it might be detrimental to their diet to give them much.