My 1894 (produced 1899) has a significant build up of handling oil (not gun oil) in the wrist and head area from handling. I would appreciate any advice on cleaning the dirt/grime from this area. I do not want to refinish, just a ca
reful cleaning and re-seal and wax.

When cleaning a gun stock, it is always helpful to know what finish is on it so you don't damage it. Since that often is not possible, I always start with the mildest cleaners. One that has worked well for me is a solution of Murphy's Oil Soap. Dry the wood well after using any water based cleaner. Cheap dollar store toothbrushes are good for scrubbing dirt in checkering.

A solution of household ammonia is great for removing oily dirt because ammonia will saponify the oils and make them water soluble. But ammonia must be quickly rinsed from the stock.

This is often one of the places where the use of boiled linseed oil is useful for gunstocks. It makes a good cleaner, but you need to remove the excess after cleaning, because BLO is a very poor stock finish. It may work even better as a cleaner when thinned with some mineral spirits. You can apply it with a rag, a toothbrush, or lightly scrubbing with very fine steel wool on a finish that isn't glossy.

Lots of people will recommend furniture cleaners such as lemon oil products. But unfortunately, many lemon oil products have little or no lemon oil, and may contain many ingredients that you didn't expect to find. Look at what is in this popular lemon oil cleaner/polish. What you do not see on the ingredients list is lemon oil:

https://www.ewg.org/guides/cleaners/3017-OLDENGLISHFurniturePolishLemon

Common isoproply or rubbing alcohol has worked well for me too. But it may damage or totally remove some stock finishes such as shellac. I bought one Baker pretty cheap at a gun show. It had a very dirty finish that looked like an alligator hide. I suspected it was refinished with shellac. I started cleaning it with alcohol and it totally removed all of the shellac and dirt on the surface, and the intact original finish was still there underneath the ugly mess. A dollars worth of rubbing alcohol and steel wool, and an hour of my time, probably added around $100.00 to the value.

When you say you want to re-oil after a careful cleaning, what sort of oil are you thinking about putting on your wood?

I use my own old-time mixture from my gramps (violin maker) containing linseed, carnauba wax, Japan dryer, a couple of other items, put on warm, exceedingly thin coats, once per week, over 8 weeks. Hand rubbed after each application. Appearance is as glass with a soft sheen and all grain filled. Key on old wood is to get it clean first while saving the character of 100 plus years. On a vintage arm I do not want a refinished or new look.

Try raw linseed oil for cleaning a stock, you may be shocked how well it works. A clean cloth and elbow grease to start, if that does not get it all off try a light hand and some 0000 steel wool in a small area. Wipe up all the excess oil with a cloth, raw linseed has no dryers so never hardens. Murphys oil soap is a good cleaner, be careful how strong you make the solution as it can lift finish as well.

This is my go to for initial cleaning of a gun stock. It is a traditional furniture cleaner of very early origins and can be found in many antique furniture restoration books, it works well and does not cost an arm and a leg to make up.
4 parts Raw Linseed oil.
1 part Turpentine. I have also used Turpentine substitute, and that also works well, though I do prefer the smell of real turpentine.
12 parts Vinegar. Malt or distilled.
Only one instruction shake the bottle well and use a generous amount on a cloth.