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There are a whole bunch of things that could be the source for the name, e.g. where we found most of their work (The Dipylon Master) or the potter with whom they worked (the Amasis Painter), a favourite theme (The Athena Painter), the Museum that ended up with the most famous thing they did (The Berlin Painter) or a notable aspect of their style. Like, say, The Eyebrow Painter.
Guess what kind of pottery the Eyebrow Painter made?
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Just excellent.
A frowning fish, painted onto a plate, surely makes for the best funerary offering.