Yemon, I have used the term "weak Banach algebra" for such things. I don't think there is a standard term, though. I vaguely recall seeing people simply call them Banach algebras (probably in some older papers when the terminology in the subject hadn't really stabilized).

(I ran into this issue when dealing with the Lipschitz algebra $Lip_0(X)$ for $X$ a complete finite diameter metric space. You really want to use Lipschitz number as the norm, even though this only makes it a weak Banach algebra. There's no real penalty for doing this, and the advantage is that it allows you to identify $X$ isometrically with the normal spectrum of $Lip_0(X)$.)

Yemon, I have used the term "weak Banach algebra" for such things. I don't think there is a standard term, though. I vaguely recall seeing people simply call them Banach algebras (probably in some older papers when the terminology in the subject hadn't really stabilized).

(I ran into this issue when dealing with the Lipschitz algebra $Lip_0(X)$ for $X$ a complete finite diameter metric space. You really want to use Lipschitz number as the norm, even though this only makes it a weak Banach algebra. There's no real penalty for doing this, and the advantage is that it allows you to identify $X$ isometrically with the normal spectrum of $Lip_0(X)$.)

Edit: I've just realized that this is what Gelfand meant by "normed ring". E.g., on the first page of his book Commutative Normed Rings (1960) he writes:

"A normed ring is a complex Banach space in which an associative multiplication is defined that is permutable with the multiplication by complex numbers, distributive with respect to addition, and continuous in each factor."

and there is a footnote which says "In another terminology, a Banach algebra."

A few pages in he proves that you can always achieve $\|xy\| \leq \|x\|\|y\|$ by renorming.