Maybe a Louis XIV, aged bust, Liard? They should be 23 mm, and a quarter is 24 mm. The problem with using diameters to ID 1600's coins, is that in many cases they are off by the official diameters by a noticeable amount.
First photo is OP's coin, rotated and cropped
This photo is an image from Google of the coin I think the OP's might be.
It doesn't look too much like a Louis XIV now, amazing what a good cleaning can do. Nice job there cleaning it. I now am leaning towards it being a William III farthing, as I said earlier. I am not completely sure on this one though, it does look to have been in circulation for quite a while. Amazing find right there, whatever it is, it is very old and cool, imagine the stories it could tell!
P.S., I was thinking George I Hibernia Halfpenny at first glance of the new photos, but noticed that the hair and bust were a little different when I compared it to an image of one.
Also looks like it could be a Hibernia Charles II Copper. I put it side by side with a William and Charles copper to compare. I noticed the top of the hair on yours really resembles the Charles II.
My vote right now is on it being a Hibernia Charles II Halfpenny. The bust and hair are very close. The back of the head doesn't seem to match the William as much and the Charles.
I would also like to add the the Charles Hibernia Halfpenny is 26mm, and the quarter you compared it to is 24mm.
Well, I did an overlay of a Charles II Halfpenny and the OP's coin in Google Earth, they seem to match up pretty good. What do you think?
Here is an animated gif of before and after the overlay, on of my screenshots is a little bit smaller than the other, sorry!