I really depends on what type of medium she enjoys. Honestly I've never found a set that I love because they are generally a 'sampler' of the mediums and not made for artist (ie the sets are expensive but the supplies can be cheap). Since I don't have a favorite set I'll just list some of my favorite brands of art supplies and if you want you could put your own set together. I would recommend getting a coupon and going to your nearest Hobby Lobby/Michael's/etc. or looking online at
Amazon or Blick for the best prices. I'll give you a lot of mediums since you didn't say if she was just interested in drawing or other mediums as well.
I like (and got when I was 14 as my first drawing set) Derwent sketching pencils. The lead is smooth (which makes shading easier). This is just an example of a small set but you can get sets of 12, 24, etc... Which is important because pencils come in different 'hardness' and depth of blackness (which really means amount of graphite left on the paper after a stroke with the pencil). You ALWAYS want an HB pencil in you set it stands for hard black and is generally considered the 'medium' pencil. I always used mine as a sort of do all tool when drawing. It looks like one pencil in this set might just be graphite (with no outside wood) which some people like for shading (but they break easily imo). You can find these at art stores and online and the sets are pretty economic for the quantity and quality you get (art supplies can get expensive) especially with a coupon. I'm sure there are better pencils out there but these are a step above your #2.
http://www.dickblick.com/products/derwent-sketching-pencils/
If she likes color I would recommend getting her Prismacolor colored pencils, they are expensive but worth it if she is serious. The color goes on smoother than something from the school supply isle and they can be blended to create intermediate colors. The smaller sets (about 12 pencils) can be bought for $10-15 but I know you can use a coupon on these at craft stores (and may be able to find them cheaper online). An example -
http://www.dickblick.com/products/prismacolor-colored-pencils/
Many sets include chalk pastels, which is one of my favorite mediums because of blending ability. I haven't bought a set in forever because they seems to last me so long but if I remember they are fairly reasonable for the price and quality you get. These are not your sidewalk chalk, they have a bit more richness of color and 'stick' (as in stick to the paper) to them. Unlike oil pastels (which I don't personally like as a medium) these are dry. The can be messy (as in your hands will be completely colorful) but I really enjoy using them. In sets these are generally the worst quality (I think).
http://www.dickblick.com/products/sargent-art-square-chalk-pastels/
Now if she is into painting... Artist quality paint can be expensive. Especially watercolor. I love watercolor and you can find small 'sketching sets' at craft stores. The ones in the stores should be about $30 - $40. I like the ones with blocks of pigment as opposed to tubes because they are (to me) easier to transport and conserve. This is an example...
http://www.dickblick.com/products/w...n-watercolors-compact-set/?clickTracking=true
Artist quality acrylic paint is also expensive and I actually sue the cheaper bottles that you can find in the craft section as opposed to the art section of the store. Mainly because I'm not that fantastic at painting and its basically what we used in high school with good results. I think the main difference is pigments used (synthetic vs natural, anything with natural pigment is more expensive). But here is an example of what studio artist use.
http://www.dickblick.com/products/blick-studio-acrylics/?clickTracking=true
Frankly the world of art and mediums is huge. And art can be an expensive hobby but quality supplies are worth it and can inspire a lifetime of creativity. If she doesn't have a good quality sketchbook she needs one (almost more than any medium, even with cheap pencils you can get decent results with good paper) I recommend Strathmore or similar, its relatively inexpensive, decent quality, and can be found in most stores. Everything I recommended but watercolor should be able to be used on a good quality paper.
http://www.strathmoreartist.com/draw-sketch.html
If you get watercolor supplies you will need watercolor paper, its thicker and has (generally) more tooth on the paper which allows for better absorption. Regular paper cannot hold up to the wetness of watercolor without ripping. Acrylic can be used on regular paper (if it is pretty thick) or canvas (which is fun, but kind of expensive) I recommend taping the paper down to a piece of board or something with masking tape so it doesn't move so much and you can get sharp boundary lines (for matting, if she ever makes something to frame). The thicker the paper (the heavier the weight) the more abuse the paper can take. If you get paint brushes they don't have to be extremely expensive, and they sell decent quality sets. Also you don't have to get the exact brands I posted here, they are just for example and (usually) I have tried and like them. The only one I haven't tried is the Blick acrylic and that brand of chalk pastel (but I know people who like it!)
I know you probably didn't ask for this much information or you may only be interested in a drawing set. But hopefully this can be of some help! I love art and encouraging your artist. I love that you are encouraging you daughter like my mom encouraged me. Maybe you could get her some art classes or she could take some at school to further discover mediums she likes!
Cheers!