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 $Lip_0(X)$.)