If that's the only stock problem, then I wouldn't restock it...then you've got a re-stocked NID...that stock can be easily repaired with a bowtie or a BX staple, inletted into the face of the stock...then the little poor inletting that's left (around the top tang) can be filled with colored acra-gel...
Looks McGraw engraved...
See the end of the screw, on the top tang, behind the safety button.? Notice the relationship to the surface of the top tang...If you alter this screw as little as 1/4 to 1/2 turn in either direction, chances are that you will have single trigger problems...so keep that in mind if you re-stock it, or just tighten and glue the original stock..
the trigger itself doesn't look like any Miller's that I've seen, but there are many Miller variations that I've not seen...
My guess is that it's a Howland trigger, maybe even an illegitmate cousin of a Miller ...that's probably too harsh of a comparison, but I believe that Ithaca was fitting their guns with single triggers long before they enjoyed patent protection for the mechanism...This is not the first time I've seen that under-cut trigger on an Ithaca either, (as others will probably tell you)...
If you have the right screwdrivers to remove the stock, and show pics, there is a 50/50 chance it could be identified. The fact that it's selective rules out Howland's second or third triggers...so logically the first comparisons would be the 1931 Howland design, and the 1925 EE Miller.
It's interesting to wonder what happened between Ithaca and EE Miller, because EE averaged a new patent trigger nearly every other year in the 20's (6 pats), then in '29 he stopped, only to start up again nearly a decade later...early in that same decade, that started out in the Depression, Howland patented at least three trigger designs...so I wonder...did EE Miller get chopped up by the Depression or did Ithaca stop using his trigger for economic reasons? Why was Ithaca pressing for their own Howland design? Did they do it because Miller was out of operation during those Depression years, or did they have a falling out with EE Miller as they did with Emil Flues?