How to politely negotiate PhD funding offer?
You can always ask, and the polite way to do it is simply to ask politely, while indicating that you have a strong interest in enrolling in the program.  The best circumstance is that you have a financially superior offer from another program which is of equal or slightly better quality.  Then the graduate program in question may see a small amount of additional funding as a reasonable expense to ensure your enrollment.  (If the other program is more than a little bit better, then everyone will be expecting you to go to the better program...unless you have personal reasons to want to go to the less good program, in which case you do not need a financial incentive.)  

user45756 "guaranteed" that such a request would not lead to summer funding.  Of course no one person could possibly make such a guarantee, and I have very occasionally seen additional funding given to graduate students to help attract them to a program or keep them in a program.  But user45756's answer is still accurate in spirit: in the overwhelming majority of cases you get what you get.  

However the expression The squeaky wheel gets the grease is very applicable to academia: people who ask for a little bit more* tend to get a little bit more.  Your request for summer funding is unlikely to magically result in a higher starting salary, but it may well result in your being placed higher on lists for various summer funding opportunities.  I think it is a good idea to say something like "I understand that additional funding may not be possible, and I very much appreciate your offer.  However, I am sincerely concerned about quality-of-life issues while enrolled as a graduate student, and I would very much appreciate being told of any other funding, scholarship or teaching opportunities that are or may become available."

*: It is true though that people who don't know the culture well enough sometimes ask for a lot more when they think they are asking for a little bit more, and that often causes them to be taken less seriously or ends negotiations.  (Imagine if you had an assistant professor job interview at a public American university and, after shaking hands with the department chair, told them that you were holding firm at a $100K starting salary.  It is more than likely that you've just talked yourself out of any possibility of a job offer.)  There is a real art to asking for something in a way which makes clear that you will be grateful for any response that you get, not that you feel absolutely entitled to getting your precise demands.  A graduate student asking for extra funding should make extra clear that they are "just asking" and will be grateful for any response they will get.  I remember one long-ago friend who was hoping that the MIT mathematics department would "get into a bidding war" with some other department of comparable quality.  Of course that didn't happen, and though he did start a PhD program somewhere quite good, I could tell from this behavior (I was a first year graduate student at the time) that he didn't quite get the academic culture.  I believe he dropped out within a year.  (And then I lost touch with him, but I am willing to guess that he now makes much more money than I do...)