Shoe Goo is really designed primarily to function as a wear surface rather than an adhesive, although obviously it has to adhere to work.
Contact glue would be better.
But all means, try glueing it in place. But if it’s just that tiny lip that holds the end of the strap down, is think it wouldn’t be very noticeable if you replaced it with a leather one. I wouldn’t go with a rubber band, that would spoil the look. But a tiny black leather loop isn’t going to look all that different from the nice reptile leather of the original.
Except for the clasp and the Hamilton stamp on the buckle. 
Wait, i thought the rest of the buckle was okay?
Sorry, even after my walk, I’m not awake yet. I thought you were suggesting replacing the original strap with a new leather one. I have started my second mug of coffee now…
If it wasn’t stapled or stitched then it was almost certainly originally closed with contact cement, probably Barge brand, and that’s the best type of glue to repair it with.
CMC, occasional, semiprofessional, leather worker.
OMG it’s almost like all the people here who are trying to answer @Johnny_L.A’s actual question about what glue he should use rather than get him to do something else are in agreement…
Unfortunately that result fulfills one of Cecil’s founding Great Prophesies, and therefore signals the End Times are nigh for SDMB.
Post quick before the lights go out forever.
Just kidding … I hope! 
You, my friend, are an optimist.
I, however, am not. It will take a little longer, I suspect.
IMO, if this is a collector’s item, bite the bullet and buy the genuine replacement. Otherwise you’re cheapening it.
IMHO I would use fabric glue. It’s meant to be pliable, and withstand motion and heat. Aleene’s is the go-to for fabric glue, and they make a leather glue. You may want to consider their fabric fusion glue instead since it is for heavy use.
Completely disagree. If the old strap is repairable to a good standard, keeping the original is superior from the collectability standard. Way too much stuff is treated as disposable these days, and analog watches in particular are intended to rather be serviced and repaired over their life.
Sure, but if it were collectible, the best option would probably be to get the keeper repaired professionally. Next best is replacement strap. DIY? Way WAY down the list.
Assuming the only issue is with the keeper or watch band loop, that is something that few people will see so I don’t think it matters much if you leave if off (perhaps using Velcro to hold the loose strap end in place) or replace it with a generic loop.
DIY by gluing up a thing that was originally glued in place, using a similar type of glue, is repairing to a professional standard.
And i actually think that may be where the OP stands right now.
Perhaps, if you have the professional’s skillset.
Well, i don’t know about @Johnny_L.A , but glueing isn’t a super-high-skill activity, and i know that i am as good at glueing things as a professional.
I’m pretty dang good with glue myself. I wouldn’t say professional. (Is there pro gluers?)
I was a professional artist and I worked with leathers in book binding. And lots of glues.
If he finds a great glue and is careful it can be done.
I can’t swear it will stay forever. But making-do is nothing to frown at. It’s in my DNA.
If the watch is truly valuable, I might fix the band, maybe buy a new leather loop keeper down thingy.
If I was then gonna sell it I’d deck it out with new band. Save the original. It has value even with a missing thingy.