Is it offensive to ask my PhD advisor for permission to record our research/technical discussions?
You need to ask first, and you need to ask without the recording device in your hand.  

But sure, why not ask?  If I were asked, I would find it a little strange at first, and then I would think about the fact that this shows a real commitment and diligence to keeping track of what is said during our meetings.  One of the biggest issues with student/advisor meetings is that it feels like they should happen very informally, but information is often being provided in a way which the student cannot easily or perfectly understand or even take in.  So there is often a lot of repetition over a period of weeks, months or years.  (Not just between students and advisors, come to think of it.)  With that in mind I would be inclined to say yes.

I also strongly disagree with another answer which suggests that an advisor might drop a student for making such a request.  This seems to contribute to a sentiment that students should be afraid to discuss things with their advisors, which I see so much of on this site and think is very unfortunate.  It would be incredibly unreasonable behavior to drop a student for making this request or really any request which has anything to do with the student's academic or professional life.  

Added: My response is not meant to apply to interactions with other people aside from the OP and the advisor.