Ash is an English name for the a-e ligature æ. It gets that name because in the Runic alphabet æsc was the name of the rune with the same sound as æ, and it got carried over.
(When æ is properly italicized in a serif font it looks unnervingly equivalent to the o-e ligature œ. Try not to let this faze you as much as it does me.)
According to OED, the original ash rune looked something like this:
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The ash rune was named after the ash tree, not the powdery remains of something burned or the Pokémon master.