I'm usually fairly cautious about which animes I introduce to my friends who have never watched any before, because it can be weird and intimidating if you're not accustomed to it. One series that has worked really well as a first anime for a couple of my female friends is Scrapped Princess (although one said she spent a lot of the series being confused....she still loved it). If she likes drama, definitely go with Monster. A lot of people that are skeptical of anime are surprised by its lack of supernatural elements and giant robots, and then quickly become hooked on the story.