I switched PhD programs after a month, do I mention this on my CV?
I think you should not bother mentioning the first (unfunded) PhD program.

From your post (emphasis added):


  After a failure of communication, I turned down other funded offers to accept an unfunded PhD. I had some money saved and tried it out anyway, but after one month quit and went with a previous offer (after telling them I'd made a bad decision)!


Since you only spent a month in this PhD program, I can't see how it is really relevant to your experience, nor will it result in a significant gap in your CV if you omit it. Moreover, you probably do not have many achievements from this month, so it is more likely to just result in unnecessary questions when people read your CV.


  I worked for 3 months specifying my skills for the defaulted PhD, do I mention the experience on my CV?


What do you mean by "specifying my skills"? This could be a good thing to mention if you were working on, e.g., online coursework, but you don't need to mention that it was for the PhD that you quit.

On the other hand, if it was just you working on skills on your own, then it won't help to mention it.


  On one hand I don't want the blip recorded, on another it leaves a little time gap between finishing MSc and entering PhD


How long is the gap of time? If it's just a few months, it will be ignored. If it is a full year, it may be something you will want to explain to someone if they ask, but I don't think it will be necessarily critical.


  and the institute name is a major buzzword for careless recruiters.


I am not sure I understand why it would be a "buzzword" -- is it a poorly regarded institute? Anyway, combining this concern together with all your other reservations, I think it would be fine to leave it off.