I'm not going to suggest any specific brand or bow, but I will suggest you look for the following: camo in color, it will resell better than anything "pink"; adjustable cam and poundage from around 17# to near 50# or more and adjustable drawlength from 22/23 to around 27/28 or more. You are looking for a youth bow or women's bow that can easily be resold to a large audience. These bows don't usually drop too much in price or value.
Just had the same question posed to me by a woman who wants to get into shooting, not necessarily hunting yet. I did some research at local shops. Here is what I would suggest:
Depending on who is a local dealer in your area and the bows they carry, then in no particular order, Hoyt Ignite Package, Bowtech Fuel Package, Mission Craze II package. Dollar for dollar, all you have to add are arrows and a release. With that, figure the total investment will be at least $475.00 for new equipment. For the $300 ball park you talked about, then you are going to have to go used and best bet right now is Archery Talk, used bows are at a low cost but going quickly.
I'd go with the Craze. Has all the adjustability needed and a decent bow for the price. But mainly for resale value. They sell on AT for close to MSRP. That way, if she becomes not interested, you don't lose as much $, or if she loves it and wants a higher end bow, the resale value is there to put toward a new bow.