Negotiating PhD stipend
I must say that I am surprised with the answers here.
I am willing to bet a fair amount of money that you will not get a raise from your institution. And I can think of a million reasons. Here are a few big ones.
A part of my reasoning stems from my personal experience, where I had multiple offers to top-tier US universities, and also from a slightly lower level US university, which offered me a salary that was almost 10k higher. When I took this offer to other top-tier US universities, they shrugged and said, "Oh well. You can't have everything." Also, in my experience, the top tier universities offered similar amounts that were not worth negotiating.
Another reason is that grad students are very replaceable. You may have some relevant experience, but they can easily fill your seat with someone else (and never know what they're missing out on, because you haven't demonstrated any of your academic and research abilities yet) so it's not really worth it to the admission committee to try and negotiate your salary.
Also, note that you may not have been admitted without your relevant work experience, so you may have already "cashed out" from your experience.
Then there's the fact that a lot of grad students are admitted each year (as opposed to a couple of professors/postdocs). If the administration tried to negotiate salaries with 20+ graduate students a year, it would be a nightmare, not to mention that it would seriously hurt the morale of the student population to compare salaries and realize that there is some sort of a ranking amongst the grad students. Being in grad school is already damaging to your ego; can you imagine the student next to you getting paid a lot higher salary, and realize that you're not as wanted as her?
Anyway, I have never heard of grad students negotiating higher pay (unless you count getting federal funding, or research support from your advisor, or extra summer teaching etc.), and combined with the fact that you've already accepted your offer, I really don't see your institution increasing your salary. In fact, I wouldn't even try, because you're already committed to going there, and the people will remember you for trying something like that (OK, maybe it wouldn't hurt to ask once, but I wouldn't try very hard).
That being said, I am aware of a few cases where the grad students had two-body problems, and the institution was willing to accommodate that, but only in the cases where at least one of the two applicants were stellar.
I believe that I should get a stipend that's higher than the minimum requirement set by the university.
In a context such as this, the word "should" carries very little meaning. To a very large extent, what "should" happen is what you have the power to negotiate.
How should I go about doing this at an american university?
(Note: I have already accepted the offer)
Unfortunately, by accepting the offer you have given up essentially all the leverage that you had. You may be able to threaten to un-accept the offer, but this would be a rather dishonorable thing to do (with potentially negative consequences for your reputation and future academic career) unless you were in a situation where the increase you are seeking to your stipend was absolutely crucial to your well-being, and would likely also be perceived as an empty threat unless you also had other offers (from PhD programs or from industry). So, I would expect that your chances to have your stipend increased at this point are very low. You might have better luck if you wait a year until you have already entered the program and demonstrated to your PI that you are an unusually talented and hardworking student and are providing their group with great value for what they are paying you; even then, your leverage would be low, but leverage isn't necessarily everything and your PI might be receptive to moral and reputational considerations, such as the PI's desire to have other people know that they are treating their students fairly.
With that said, I believe asking for pay commensurate with your skills is a very reasonable and acceptable thing to do, so it can't hurt to try, and can pay off (either by having the request succeed, or failing that, simply by signaling that you are a confident person who knows his worth, which may earn you a small bit of respect) if done in a professional manner. The way to do it would be to write an email (phone, skype or a face to face meeting would also work obviously) to the graduate program director communicating your request politely and including any relevant information on how much money you were earning, how much other similar positions pay (specifically, if you can get information showing that other PhD positions at universities with a similar profile to yours pay more than what you have been offered to people with your level of experience, that could be helpful), etc. The better prepared and explained your request is, the higher the chances of success, though as I said I think they are still low. Avoid formulating your request in emotional, accusatory or drama-laden tones, or implying that you have some moral entitlement to being paid more. And make sure you get the details right -- asking for a stipend that's considerably higher than what your experience suggests is reasonable will only annoy the people you are trying to negotiate with.
Finally, I should add that I don't have any specific experience of incoming PhD students negotiating higher pay, but I have certainly heard of departments improving the terms of an offer to students they want to attract (typically before they have accepted the offer and when they have competing offers, alas). Note that in many departments the pay is standard and dictated by university policy or various union-negotiated labor contracts that are fairly rigid and do not allow much room for paying different students different amounts. However, even in such cases where the department can't pay you more there should still be some flexibility, in the sense that if they really want to help they may be able to offer other improved terms (such as access to funds for traveling or buying equipment, or replacing a teaching assistantship with a research assistantship).
Good luck!
I agree with Sana.
I don't think PhD stipends are negotiable. In the US, the grant of a research group is actually controlled by the university. Every time the PI wants to spend it on something, (s)he has to make a request. In most case, these requests are just formality, and are always approved. But I doubt that (s)he can request for his/her PhD student a salary which is higher than the norm of the university.
I think this is also the case for postdocs, and I'm not aware of any PhD students/postdocs who can negotiate their salary.