Today, I've heard someone speak of a basis (of an ideal), meaning a generating set. All the time, I was fine with the term Gröbner-basis, but when it comes without the prefix, it's a bit funny, since basis, morally, is earmarked for something generating freely.
So I wondered which term was used first, and how, as well as by whom of course. The first thing that came to my mind was Hilbert's basis theorem, but Hilbert did not speak about bases. The next names to consider were Gröbner and Buchberger. And in fact, Gröbner used Basis for generating systems of ideals whereas speaking about a module, a basis was supposed to generate freely. (See, e.g., Moderne Algebraische Geometrie, 1949, Springer Wien & Innsbruck.) Not surprisingly, being a student of Gröbner, Buchberger also called generating sets of ideals bases.
I realise that a full description of the evolution of basis could be too much to ask for, so I would already be happy to read an answer which builds a bridge between Hilbert's basis theorem (Über die Theorie der algebraischen Formen, 1890) and Gröbner's terminology, possibly by pointing at the first one to refer to it as basis theorem.