I think this is really going to depend on a number of circumstances. If you can provide more information, I think it would really help in understanding the situation a bit better.
I live in a very popular destination and vacation spot in Europe. Needless to say, I have lots and lots of people who would like to come and visit. There are so many, in fact, that although I would love to be able to be a perfect host to all of them, the number of visitors in a year really can get out of control.
I live in a studio apartment with my boyfriend. While we do have a very comfortable and large guest mattress, in addition to a couch, this means that we don't have a separate guest bedroom...or any privacy when guests are here. Any of my nearest and dearest friends are more than welcome to come stay with us at anytime. I love them, I want to see them, and I want to spend as much time with them as possible.
However, I also get requests (or very, very obvious hints) from people with whom I'm not especially close, from friends of friends (including people whom I don't actually know), and from others who are wealthy enough to make them more than capable of paying for their own hotels. It's not particularly nice if you get the feeling that you're kind of being used to give someone else a free vacation, especially if you are a good host and will be spending a potentially significant sum to host them properly, which is the only way I want to host in the first place.
To complicate things even more on our side, our business is very close to our apartment, and I wind up doing a lot of work from home (in that one large room). This can make things more difficult than they might otherwise be.
Only you can know how close (or welcome) you are with this friend and how welcome you might be to stay in this person's home.
Also, will you be traveling alone, or will you have friends with you?
Dontknownuthin hit the nail on the head. You need to call you friend, chat like friends do, and then you can tell her that you're planning a trip to her city. Let her know why you want to go. Ask if she can recommend any cheap hostels in the area. If she offers for you to stay with her, great. If she recommends a hostel, well, you've got your answer.
Personally, unless you're really close, I don't think it's in great taste to just ask if you can stay with the person flat out. They should have the opportunity to invite you. It might seem silly, but it can make a difference regarding how the host will feel about it.
By the way, if you do stay with her, I really do recommend that you bring a nice gift upon your arrival and/or treat her for lunch or dinner during your time there. It might seem small, but this will make a difference long term, especially if you want to go out to visit her on other occasions.
Honestly, if it's just a night or two, it shouldn't be a problem. Just make your plans clear. I've had people tell me the same thing, and they wound up staying for a week or two instead. Believe it or not, this has happened to me on more than one occasion.
If she doesn't invite you, don't hate her for it. I'm sure she has her reasons. Just meet up during your time there and enjoy your time together. If your relationship grows, she might have you stay with her the next time you're in town. By the way, hostels are a travel net that anyone should be able to afford. They're not fancy, but they'll do in a pinch.
Promise to let us know how things go! You've got my curiosity going.