Should I give authorship to a senior person whose main role was to connect me to other people?
It is more typical for me to weigh in on the side of not putting senior people's names on papers when their contributions were borderline or less.  However in this case it seems to me that it probably would be a good idea to include your former postdoctoral mentor as an author.  


  Her role was to connect me with the other people (she knew them, and they had access to a specific patient population I needed).


That is a contribution.  It sounds like it is not an intellectual contribution, but in signing up with a mentor you probably were hoping to get increased access from her connections...and you did. 


  Also, my mentor and I talked about some related ideas years back, but never the specific experiment that the paper is about.


That does sound like an intellectual contribution.  You didn't talk about the specific experiment, but that doesn't mean that your discussion didn't contribute to the design of the specific experiment.  In my opinion high-level general discussions can be at least as valuable as specific implementations...especially when they lead to specific implementations.  


  She thinks the experiment was her idea, but she is mistaken about this: when I first told her about it, it was already completely designed, she gave no input.


Well, you are giving your side of this, and you are aware that her side is different.  Do you have records that would support your version of events?  If so, I would certainly discuss them with your mentor before deciding not to include her on the paper.  If not: if the only thing you have to rely on is your own memory of events, then even if we grant that you and not she are remembering correctly, you still may want or need to act in a way that an external observer would regard as correct.

In terms of your pros and cons: as you mention, keeping a good relationship with your former mentor is a real pro.  You seem to be able to put yourself in her shoes well enough to understand why she would feel slighted and react badly, so in my opinion you need to not let that happen, either by including her as a coauthor or by resolving the different perspectives you have on the situation.

Your cons are not very persuasive to me.  As you say, the paper already has five authors.  I don't know about you, but with that many authors I am going to assume that some have rather minor contributions unless I am specifically told otherwise.  I don't understand how adding a sixth author undermines the independence of your contribution (presumably you will be first author).  You mention that your mentor made no scientific contribution, but that's not clear to me since, as above, you mentioned earlier conversations on the general topic that led to your specific experiment.  I also think that people should not be added as authors for no intellectual contribution...but this is not a very clear case for this kind of principled stand.