In attempting to sweeten an acceptable job offer, the best approach is to play upon the ego and power of your new manager. Not in a negative way, but giving him/her a chance to show his/her strength within the company. You need to realize that you are in one of the strongest political positions you may ever be in with your new employer. Utilizing this technique can have the side effect of enhancing your future power within the organization—if used correctly.
The key to using this technique is to overtly empower your new manager to work on your behalf. You give her/him power in two ways: (1) by accepting (yes, I said "accepting") the position, you give her/him power because s/he has added the desired person to her/his team; and (2) by asking her/him for her/his assistance in meeting your further needs, you give her/him an opportunity to show her/his power within the organization. How to do this? Consider the following example:
"_____ (name of boss), I'm calling you with some very good news. I would like to accept your offer and I'm looking forward to working with you and becoming a valuable member of the team. (Wait for their positive response.) I am committed to working with you, and as my future boss there is (are) a (two, three, some) minor issue(s) about the offer that I want to make you aware of. I don't know if you're able to make changes in this (these) area(s), but I'd surely appreciate your looking into that possibility. Namely, would it be possible to _____ (specifically detail your requested changes)?"
The beauty of this technique is that it provides two things. First, it locks in your acceptance of the job and takes that acceptance out of the negotiating. Second, it leaves open additional concessions that may be given to you at no cost. Please note that most books and articles on negotiating would argue against giving away your greatest negotiating chip—acceptance of the position. But this argument is correct only if you can afford to lose this chip. If that's the case and you truly want to "spin the wheel," you can use the Unacceptable Offer Negotiation Technique. Yet you risk losing the job offer entirely if you choose that approach.
By closing out the offer acceptance portion, you have locked down the one thing you cannot afford to lose, yet you leave open other areas. Your retained negotiating chip? The hoped "show of power" on the part of your future boss. Even if your boss is not able to deliver, the fact that you turned over your personal request to your boss will make him/her more willing to help you in any future needs. Even if all you get is exactly what was originally offered, you are given by default a future negotiating chip.
Why does this technique work? Simple. It's a win/win situation. The key to successful negotiation is that each side should stand to benefit. In this example, the worst case scenario is that you will have the new job you want and the company will have the new employee it wants. The best case scenario is that your new boss will feel personally and professionally satisfied because of her/his ability to immediately enhance your new position by meeting your additional requests. Even if your boss is not able to provide any (or all) of the requested concessions, s/he will feel privileged that you brought her/him this issue to handle, and will then feel an obligation (due to guilt?) to help you in any future requests.
The greatest benefit of this negotiation technique is that it has zero risk (by securing the job offer acceptance), yet invariably persuades most managers to give in to some or all of the concession requests. Why? Plain and simple: ego. Your new manager wants to show that s/he has the power to make things happen. At this point, you are both on the same side, working for the same goal. Later in your career your eventual goals may conflict with your manager's, but at this time, if you can convince your manager of the value of your request, you will have a strong advocate on your side.
So if there are areas of the offer you are seeking to redefine without risking the offer itself, this is a safe, effective way to test the waters. The risk is minimal, while the payback can potentially include important changes. It should be noted that this technique cannot be used for significant offer changes. If you need to make changes of great magnitude, you will probably need to follow through with the Unacceptable Offer Negotiation Technique.
Good luck!