Thank you for all your advice. This is very helpful. Can you clarify what you mean by PA? How is this different than a TA or RA?I don't know specifically how things would go down and I get the feeling that it would be different based on the school and the culture. I think you would make a stronger case if you came in with a different offer from another school.
Alright if I were you, here's what I would do. I would talk to your point of contact person at the school you are trying to negotiate with. Tell them frankly that you are very interested in their program and that you would prefer them however you have another offer from school X. Let them know that school X is offering $Y and that you were wondering if there were anything that their school can do in regard to your stipend or TA hours. Ask if there is a fellowship you may be qualified for, see if you can negotiate working as a PA (or whatever the school calls it) for your adviser rather than a TA. Whatever you negotiate, ensure that you get it in writing and that the timeline is firm. Basically you don't want to get reduced hours for year 1 only to get slammed the following years. I would also ask maybe about summer support. Then also ask about maybe getting more money for conferences etc...
In general there is not a lot of flexibility in terms of real money that schools can offer. Generally that is set by University. But yeah, reduced hours, fellowships, scholarships and conference funding are things that the school may be able to move on. Here's the thing, though, you need to go into this not as a business person negotiating a salary increase (save that for when you're on the market and get a job offer). Go into this as a humble student just trying to make ends meet. There are a few things that you don't want to do. First, upset the department and amake them think that you are difficult to work with. They hold your future literally in their hands. Don't make enemies from the start. Second, don't talk to your fellow classmates about any of the things you negotiate. It will only lead to jealousy and won't make you any friends. Trust me, you'll be seeing those classmates more than your family or friends for the next 5 years, don't start off on a bad foot. Finally, I wouldn't ask the school to match another offer, but rather ask if there is anything that they can do to increase your funding or lessen your load. Framing it in that way is a little better. One more thing, if you negotiate stuff with the school, you better accept their offer. It would be a terrible thing to do to a school to go somewhere else afterward.
May I send you a PM for a different question?