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Wood glue, always. Sandpaper the dowel and hole to give a better surface for the glue to hold.

Wood glue works well; when cured it can flex a little as can wood itself. Epoxies tend to be rather brittle.

Epoxies vary a lot: slower epoxies (e.g. Araldite Precision) are (i) much less exothermic, and (ii) liquid for long enough to soak in a little. They need the parts to be held stably together during curing. They should be stronger than fast epoxies in this application, but they're still not as good as wood glue.

The key to a repair like this though is to pack the hole well, so you're not relying on the glue to take the weight, just to keep the joint stable.

Wood glue, but not for the reason why you ask. In terms of long-term stability, the observable difference is exactly zero. There is no force acting on the dowel, and there is no exposure to water to be expected.

You could probably stick the dowel in with a bit of spit and dirt, and it would do. Still wood glue is the correct thing to use.

Wood glue is non-toxic, easy to handle, and it works very well for wood. In fact, in absence of water, it is one of the best choices for wood (unsurprisingly) rivalled only by polyurethane (which is much more expensive and somewhat on the toxic side). Wood joined with wood glue, even when done with bad skill in unfavorable conditions, usually holds forever, or until the wood breaks to pieces, with the glue still intact and wood splinters protruding from it.

Epoxy, on the other hand, is vile chemical stuff that certainly has its merits where you need it (above all, it's waterproof and insensitive to most chemicals, plus it can also be used to "fill holes", give smooth surfaces, etc.), but it is uncomfortably sticky shit that spoils everything and anything it comes in contact with, and by no means as easy to handle as "open screw lid, use, close screw lid". Plus, it's poisonous, or at least skin-irritating. Depending on what type of epoxy you use, it may create poisonous fumes while curing (or significant amounts of heat). There exist "epoxy" types that are in fact not epoxy at all, they only look the like, and aren't necessarily easily identifiable by a layman. Not all epoxys even cure the same (there's sticky and non-sticky ones). You must get the mixture right, or it will not work well (cure more slowly, or inconsistenly, or not at all, or something different).

Wood glue, you can wipe off cleanly without leaving a trace using a piece of cloth, should a few drops fall to the floor. It's something you can wash out without much hassle (unless you let it dry) if it stains your clothes. It's something that will not provoke a skin reaction on your fingers or take half an hour to get clean fingers which are able to touch something without sticking to it, again. It is something that your three-year-old could, in the worst case, eat -- and most likely not a lot of terrible stuff would happen.

For installing a dowel in the hole I'd prefer wood glue. However, given that the objective is not to plug an unwanted hole with a dowel but rather to repair a screw hole...

The hole could be filled with epoxy alone - no filler wood. The hole is drilled out oversize and then filled with just epoxy. An interesting outcome of the epoxy solution is that the hole is left with threads that fit the screw perfectly without any further compression or deformation of the surrounding wood.

The epoxy solution doesn't have to hold the screw permanently. If the screw is coated with some kind of mold release like wax before assembly then the epoxy will fill around the screw but won't bond to it. Alternatively, if the epoxy is bonded to the screw, the bond can be broken by heating the screw with a soldering or wood burning iron.

There are some descriptions at the West System web site.

A big advantage of the wood fill method is that the hardware and screw are installed later after the glue is dry. This avoids the clamping hassle that's inherent with the epoxy method.

You don't want to dowel the stripped screw hole: to plug it instead. The difference is that a plug is cross-grain while a dowel is long-grain.

The reasons?

Using a dowel...

• Gives you end grain to screw into. Screws do not hold well in end grain.


• Gives you a short- to long-grain glue joint. Such joints do not hold well regardless of the glue used.

Plugging

• Long grain to long grain joint. Nearly as strong as the original wood.


• Screw goes into long grain. Much stronger than a screw into end grain.

Procedure

• Get a plug cutter of suitable size and cut a plug from the same kind of wood.


• Drill out the stripped out screw hole using a bit the same size as the plug.


• Dip the plug in wood glue (which see below).


• Seat the plug and align its grain with the piece being repaired.


• Tap the plug home — gently.


• Let the glue dry.


• Trim the plug flush using a sharp chisel or block plane.


• Drill the new pilot hole using a bit of suitable size for the screw in question.

Done right, this sort of repair is virtually invisible, and, nearly as strong as the original wood.

Glue it in with a decent wood glue. Glue choice should be determined by usage: Epoxy is the glue of choice if the repair is likely to be wet. However, epoxy is also non-reversable, and takes a long time to cure. If it's ordinary furniture or cabinetry, I would suggest using either Titebond or liquid hide glue.

There's a simpler way, though. Don't plug it if you don't have too: just move up one screw size. If it's a #10 wood screw, move to a #12.

If it,s a quick-n-dirty, wood glue.

Otherwise a competent epoxy, period, becuase you can dial it in to be exactly what you need.

I am not talking about random hardware store packs (not least, most are far too fast). I'm talking about a well-developed epoxy system such as West System. However, we could fan this out into a product like Git-Rot if wood deterioration warrants this. They will play well together; we overcoat West System, Git-Rot and Abatron all the time.

You are modifying the epoxy in two separate areas: Bonding filler, to allow you to reliably bridge irregular gaps between surfaces; and as general filler, if you need to fill inaccesible voids, or simply want to use the epoxy as the fill material instead of setting a dowel. As a filler, the epoxy can be "dialed down" to the same softness as the substrate, and that allows direct screwing the same way you would the substrate. The fillers also assure good characteristics from the epoxy.

You could also mix the epoxy to be harder and make it more like drilling into a hardwood bos, but for that matter, you could just glue in a threaded metal sleeve and have it act like metal.

Neither is really needed. Use a slightly over-size dowel and hammer it in there. Sand off or cut the excess. It won't come out.

(edit) caveat: Obviously this answer is only applicable where the recipient wood is substantial enough to take a tight fitting dowel without splitting (example a door frame with stripped screws). If not, I recommend an exact fitting dowel (drill out to fit if needed) and a small amount of wood glue.