Since the house is 100 years old, the stairs are probably constructed using a tapered groove and wedge system. The stringers typically had tapered dados cut into them to receive the stair treads. Wedges were tapped in with glue underneath the tread to lock the tread into place.
Check if this is the case by looking under the stairs. Hopefully the underside is accessible, if not you might have to remove some drywall or plaster.
If the stairs are constructed using tapered dados, please don't nail or screw the treads in to stop the squeak. It might work for a year or two, but the treads will expand and contract, loosening the fasteners which will then squeak worse than ever.
Assuming you can get to the underside, find the squeaky stairs. If the wedges are still there and sound, remove them, clean them, apply glue to the faces and tap them back into place with a hammer, tightening up the stair tread. If, as is often true, the wedges are missing or splintered/broken, cut a new wedge using an existing wedge as a model.
My first thought is a dowel, glued in place. This will stiffen it some, but leave the crack. And epoxy can be on the brittle side.
So next, I thought, why not use a metal dowel? It adds more stiffness, more strength, but you still have the crack.
Another idea was a piece of all thread rod. Screw it right into the hole. This will give a great deal of holding power, preventing the crack from widening. But still you have that crack to fill.
Finally, I realized the perfect solution. McFeeleys sells it, in the form of a LONG screw, that is threaded only on the bottom half. Pre-drill a hole to get it started, and countersink it, so you can put a plug in afterwards to hide it completely. Then use the threads to draw the crack together as the screw tightens down. Add some glue (or epoxy if you prefer) into the crack before this operation, and the board will stay solid forever.
You can buy these screws in various lengths as needed.
https://www.mcfeelys.com/1-4-in-x-6-1-2-fastenmaster-headlok-screws-w-spider-drive-bit-qty-250.html
Best Answer
Call the installer, builder or manufacturer for any warranty type of repair...I'm kidding. Your best bet is to get a Carpenter or Stair Builder that uses glues, screws & bolts behind or even upon the finished surfaces...finish screws are wonderful.
Nails, spikes & staples are the norm & they can't & won't last, they're held by friction alone. Glue Coated nails are only good for a decade at the most, glue only lasts if its hold can't flex or break loose. A stair is a moving part, if given the chance, & will only continue to loosen until another object retains it.