I would absolutely recommend Tung Oil! MAKE SURE it is 100% PURE Tung oil and not mixed with anything. Hopes Tung oil is one brand I have and have been using for 50 years - it's excellent. Apply 6-8 thin coats with a very light sanding in between coats with 0000 steel wool. Allow 24 hours between coats. Tung oil completely dries - unlike Linseed oil. Tung oil is also non toxic and can be used inside the kitchen. It NEVER goes bad - will last for 100 years as long as the can is closed.
If down the road you feel you want to spruce up the grips again, no need to strip. Just a light steel wool sanding and apply a few more coats. Leave alone on last coat for gloss finish. Lightly 0000 steel wool buff for satin finish. Tung oil is what I used on carry guns for decades. Works excellent!
I read the above post and something was not right. Tung oil is NOT gooey, sticky or soft -- dries fast and hard. Wither it was contaminated, mixed with something or not applied correctly. I have used Tung oil hundreds of times both personally and professionally - it has never failed me. Hopes brand 100% pure Tung Oil is what you want!