I know how she feels- I have shaved my arms in the past. It doesn't make the hair thicker or darker, and it lasts for weeks unlike legs. However, if you're okay with it you could also take her to get them waxed- if you wax arms it grows back thinner and lighter, and if you continue to wax it will get lighter and thinner each time (waxing damages hair folicles, causing some hairs not to grow back). And it's not something that's really high maintainance either, most people only have to do it every few months or so (maybe monthly at the most). I think this would take care of the problem more for her and help boost her confidence, and while I think it's important for kids not to worry too much about their appearance, I think that most of us would be hypocrtical if we told her to just deal with it.
Again, with shaving, maybe it's just me but I have never had a problem with itching or rash, and I have fairly sensitive skin. And it really doesn't grow back courser (that's an old wives tail because when it first comes back it FEELS courser, because all of the hair comes back at once and the same length). But I would still say waxing is the way to go- I did it when I got married and it isn't terribly painful, but I couldn't tell you how much it costs because my mom (a cosmotologist) did it.