PhD job seeker asked if their is negotiation around a TT job offer, and if so how it is done. This was my experience.
After I received my offers, I was given 2 weeks to accept or decline. During that time, at least one University flew spouse and I out so spouse could see the area upon request. One of the realities of a TT job search is that the candidate applies for many positions, likes some more than others, and is constrained by the varying schedules of different departments. As a candidate, you need to look out for your own interests, because no one else will.
In my case, one department was hiring for multiple positions. They met promptly after my interview, and I received an offer in less than a week (my first!). I was still interviewing at other places, and had at least one department that I was more interested in show an interest. As soon as I had the written offer in hand, I notified all the departments I was still interested in that I had an offer to try to speed up their time tables. In one case, I got an expedited interview. In another, the department still had one more candidate to interview, but I was currently their top choice. They gave me a firm date on which they could give me a decision. A third department had expressed verbal interest, but was dragging their feet on a written offer. My existing offer put their feet to the fire so to speak, and greatly sped up the process.
The offer I had in hand was good. I would have taken it happily if I hadn't decided I would strongly prefer Prodigal U. To give Prodigal U more time, I asked for another week to decide, while negotiating my salary and startup package for the offer I had in hand. Chaining offers like this is a fact of TT job search life, if a candidate is lucky enough to have more than one offer. Some people think it is unethical, but I see nothing wrong with it as long as you would actually take the job. It is a waste of time to negotiate and stall if there is no way you would take the offer. This can make the department lose out on their 2nd and 3rd choice candidate (if they in fact exist--in our searches, that hasn't always been the case). If I thought a job at Offer U was worse than my job at National Lab, I would have turned it down outright.
I eventually got an offer from Prodigal U. At this point, I had declined one offer and had another offer in hand (for more money!). This greatly strengthened my negotiating position, as did the fact that I already had a permanent job I liked. However, even if you receive 1 offer and your postdoc funding ends tomorrow, it is worth negotiating, especially for things that will help you be successful. Keep in mind, though, that a counteroffer technically declines the original offer, so be careful!
I asked for a much higher salary than my initial offer, which was scary (see above statement about counteroffers!). To justify the increased salary, I compared my current salary to the offer, I mentioned my higher offer elsewhere (but did not ask for a match--I definitely did not want to give the impression that I was all about the money, and my other offer was in an Engineering department, which generally pay better than a Science department like the one at Prodigal U). I also brought up my track record in competing for funding relative to a typical fresh postdoc (that's where being at a National Lab was a huge benefit). I was able to negotiate my starting salary up 15%, which is a pretty big deal, and definitely due to the fact that I was more experienced than a typical new TT hire. All future raises come as a percentage of base salary, so it is important to start out at a decent salary. I did take a pay cut from National Lab, though.
The major things I negotiated for were actually not salary related at all. Some other things I asked for: extra startup funds to cover user fees until I could buy my own equipment, 3 years of summer salary, support for 2 students for 2 years, doing my teaching in the Fall (when I would be ordering stuff) rather than in the Spring (when all the stuff would hopefully arrive) my first year, separate sample prep and instrument space, a spot in on campus daycare, a specific instrument lab location due to environmental issues impacting measurements, etc. This depends on your research needs. It was nerve-wracking to ask for stuff, and I hated it. I kept it professional and provided justifications for the things I was asking for, rather than trying to extract every last drop from my new department. Some things I got (extra startup for user fees, wet/dry labs), some things I didn't (daycare spot).
Something to remember is that in the cases I saw, the Chair was negotiating with the Dean on my behalf. It is to the Chair's and department's benefit to have the new TT member be successful. I got some good advice on things to ask for from some of the places I was interviewing at/negotiating with!