If she is okay with the size, a Nikon V1 for $299 is by far the best value camera that can do that trick (snap full-rez photos while recording), unlike other cameras that can only snap reduced-rez photos while recording. Just because you can swap lenses doesn't mean you have to. She also gets a viewfinder to help with composition under difficult circumstances such as glare on the LCD on a sunny day, super-fast autofocus, etc. The larger sensor size compared to most compacts will allow things like out of focus blur. Downside is that the kit lens is effectively 27-81mm, though you could always crop, to some extent (10MP doesn't give you that much room to crop if you want to print large posters though).
If she is not okay with the size or wants more reach, a Sony RX100 produces slightly higher image quality than a Nikon 1, has an effective focal length of 28-100mm, and the lens is quite bright on the wide end, at f/1.8-4.9. Since it's 20 MP, she has plenty of room to crop to make the effective focal length even higher. Like the Nikon 1 it can also take high-rez photos while recording video. Downsides: expensive, and autofocus and boot-up time is fast for a compact but can't touch a Nikon 1 or DSLR, especially for tracking moving objects.