It looks like the term ‘ideal-supporting algebra’ was written by me and survived slightly more than a decade on Wikipedia without being altered. (Well, somebody added a hyphen, a change that I agree with.) Since I put brackets around it, I'm sure that I must have heard the term somewhere, but I couldn't tell you now. Now that I think of it, a more precise term would be ‘ideal-supporting variety [of algebras]’.
And if I search for that phrase, I find it in Eric Schechter's 1996 Handbook of Analysis and its Foundations (which for some reason Google Books has classified under Business & Economics). Since I was reading this book a lot a decade ago, that's probably what I meant all along. Shechter often invented terminology for his book, when terminology in the literature was inconsistent or missing, so I wouldn't be surprised if it's essentially unique to him.
At this point, probably the best thing for me to do is to edit Wikipedia for a little bit, finishing what I started in 2002.