There's a lot of options.
First a varnish is a type of finish that can be formulated from an oil or polyurethane, or both. It can vary in viscosity and be as thin as water to make what's called a penetrating or wiping varnish. Add a pigment to it, and it becomes penetrating stain, like Minwax.
After years of using linseed oil formulations, such a Tru-Oil, I've switched to a product from Laurel Mountain Forge called Permalyn.
It's a proprietary blend of oil, urethane and other goodies. It comes in a stock finish and stock sealer versions. The stock sealer is just a water thin version of stock finish, that penetrates into the wood fibers.
I much prefer using the sealer, as it gives you a lot of control on the level of sheen that you want by the number of applications applied. With the urethane in it, once fully dried, it can be buffed up to a gloss if desired. It's very durable and even though it has urethane in it, it doesn't look like it.
I apply enough coats to get a matte sheen. It's easy to work with, but you have to have patience, as is takes more time and effort using the thin sealer as a final finish, applying multiple applications to vary the result and obtain the look that you want.
Product info:
Laurel Mountain Forge
Good info here in the instruction sheet:
Laurel Mountain Forge: Permalyn Sealer Instructions
Pics won't do it justice, this is a grip I refinished with the Permalyn Sealer, and then applied wax.