Why would a Church Parish record of baptisms from Wexford, Ireland in the 1690’s use umlauts over the letter “y” in names like “Murphy” and “Meyler”?

I’m not familiar with that convention, but I’m not surprised. If cursive is a bunch of squiggles, then dots help you tell the difference between, say, <in> and <m>. Same applies for ÿ: helps differentiate, for example, a final Latin <ii> (which could have been spelled <ij>) from a y.

At a guess.

Upsilon in Mediaeval Greek had a diaeresis for the same reason.

EDIT: This post Google Groups corroborates the use of <ÿ> to differentiate it from <g> in German handwriting.