I also have the name Ibbetson or Ibbotson as a middle name on my tree, and like yourself I have never been able to trace where it came from. John Ibbetson Widdop was born 1856 at Manningham, Yorkshire. His father was Henry Widdop b 1809 Manningham, and before that there was Joseph Widdop b 1789. When searching for Ibbetson, I usually also try Ibbotson, because I think there have been mis-spellings the further back you go.
You mentioned copperplate writing. The odd thing is that John Ibbetson Widdop was at one time, (according to the census) an Engraver/Artist. In fact, he signed his middle initial ("I" for Ibbetson) with a very artistic-looking flourish, which makes me even more certain he was an artist. To confuse matters, the "I" when he signs his name looks more like like a "J", and this means that his records sometimes came under the name John J Widdop instead of John I Widdop. ...Maybe we will never know what the correct spelling of Ibbetson is, but in my tree line the most recent spelling of it is Ibbetson, so I have settled for that.
I just wonder if there is a line of Ibbetsons or Ibbotsons who were artists or engravers, and if yours is perhaps connected to mine at some point away back in the 1800's.