I agree staying in that budget will be difficult because it requires so much more than just a lathe. You can stay budget minded though.
Id reccomend this lathe...
http://www.harborfreight.com/5-speed-bench-top-wood-lathe-65345.html
No, its not a top of the line model but its a decent beginner lathe. It gets excellent reviews and parts are readily available through many places such as Grizzly. This lathe is sold under many names but its cheaper at HF than anywhere else. Use a 25% off coupon and it makes it right about $150 bucks. Thats hard to beat. (Ill send you a coupon if you need one) Ive own this lathe and have been turning on it for 2 years without any problems. Yes, I would like a much bigger lathe now but it is perfect for turning pens and stoppers and other small things. It will be great for an 11 year old and should serve her well for awhile. I also turn pepper mills and other items allthough that requires a bed extension.
Id reccomend this tool set...
http://www.harborfreight.com/8-piece-high-speed-steel-wood-lathe-chisel-set-69723.html
They sell a cheaper set but I believe they are carbon steel. This set is HSS and is a good bang for the buck. I have upgraded most of these tools now but still find myself using these regularly. Many people will tell you not to buy a set but rather buy individual tools as you need them. Thats true to a point but with a coupon this entire set is about 40 bucks. You will need a way to sharpen these tools but I used them for several months without sharpening before I set up a sharpening station. They will hold an edge for awhile if your just turning pen blanks. Just dont let this go for too long because dull tools are a safety hazard. Hooking up with RusDemka or one of the other guys that make carbide tools is another option. I turn my pens with a set of Rus carbides almost exclusively.
Now you have about 100 bucks left to buy a pen mandrel, some drill bits, some kits, etc. Id start her on slim lines. They are cheap and they only require a 7mm bit, also cheap. Assembling pens can be done with a clamp so no need to break your budget buying a pen press. If you have a drill press Ill show you how to make a pen press jig for it and youll never need a pen press.
When you get serious about getting her into turning, send me a PM. My mom works at a safety supply warehouse. I buy lots of overstock items and items with damaged packaging, etc. Ill send you a brand new face shield, safety glasses, dust masks. I think I still have a new respirater too, if so Ill send that, all free of charge, you just pay the shipping. Ill fill the voids in the box with pen blanks!
Thanks for introducing a kid to the wonderful world of turning!