So you find yourself in a bit of a fix. You have an old home with a bunch of doors that have a non-standard bore hole measurement. What can you do? Unfortunately, none of the manufacturers these days are making door knobs or levers that will accommodate bore holes of that size. They all have adopted the standard 2 1/8" diameter bore hole and there just aren't enough people out there that are replacing the 1 3/4" bore hole size to justify manufacturing and stocking the inventory to fill that small need. So far this article isn't helping you feel any better about this situation does it?
Your first option and probably the least expensive is to re-bore your doors to accommodate the modern styles. To do this you will either need to make, or purchase a jig/template. To make your bore holes larger you'll need a large hole saw type drill bit. However, since your doors already have bore holes, there is no way to keep that hole saw centered where it needs to be. This is where the jig or template comes in. You can buy an inexpensive plastic template for prepping doors for the standard 2 1/8" bore hole and 1" cross bore. Most hardware stores will have these jig kits and they run about $20-30 USD. These templates slide over the edge of your door and guide your drill bit to keep it straight. Once you have re-bored your doors, your options for replacement doorknobs or door levers are quite expansive. You may also want to check the thickness of your doors before going to the work of re-boring them. Sometimes these older doors with smaller bores, will also be thin. Most door locks are meant to fit a standard door thickness as well which is 1 3/8"-1 3/4". If your doors are thicker than this, you'll want to check out Emtek Door Hardware which has styles that will fit up to 2 1/2" thick.
If you find you have extra thin doors, and you've just kind of had it with the old doors anyway, your second option would be to just replace all of your doors with new ones that are prepped for the standard measurements. Keep in mind though, if you are replacing old thin doors, you will likely need to replace your jambs as well which means you'll have to remove and replace your current casings.