Most Helpful Guy
I think shy girls worry about the what-if possibilities with dates, like "what if he tries to get his hands all over me on the first date." A daytime date feels safer for most girls. Have a zoo in your area? Most girls love animals.
Ask her to do something that requires her to help you with something, like "I need to buy some clothes but I'm no good at it. Would you go the mall with me and help?" That brings people out of their shells.