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Avon is most likely an English/Old English derivative of the Welsh word 'afon' which means river. Lea (or leah) is an Old English word (or Middle English word 'legh') meaning a clearing/meadow/pasture/open grassland. You can see it demonstrated in names like Hayley, Ashley and Leigh.

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So, I'm guessing that Lucy Maud Montgomery was giving her fictional town a name that meant something like 'river by the meadow' or 'river clearing'. And the name Lynn means 'lake'.