Filling an old hole to offset it a few mm and put in a fastener, what is the best approach?

Use a hardwood dowel and PVA wood glue. I mean, you could use hide glue if you want, but don't use epoxy. We want something that is strong and workable like wood. Epoxy will be hard, but it won't be easy to work. And the strength here is about a tight glue joint, not super hard cured glue. We want it to be able to saw, drill, and scrape like wood.

In my opinion, typical wood filler isn't going to hold the head of a torqued down screw very well, and it does not really adhere well to the surrounding wood. (Perhaps there are some more exotic types of filler, but then we are getting away from what should be a very simple, time-tested, and common fix: wood glue and a dowel.)

Slightly shape and chamfer a dowel so it goes into the hole snugly. You want a nice tight interference fit, but not so tight it pushes all the glue out.

Install it with enough glue to evenly wet all the mating surfaces (use a small paintbrush if necessary) with only a little squeeze-out, and let set. Once the glue is set to a workable amount (see the description on your glue bottle) flush cut the proud part of the dowel, and scrape flush if necessary.

As for fasteners, it depends on how this hole is being used. If it is intended to give access for a wood screw to go through and fasten to wood behind it, you drill out the hole to the dimensions of the threads. The idea with wood screws is that the head exerts the force on the top piece (with few, if any, threads engaging with the top material) drawing the two pieces together.

I am looking for most structurally sound solution

And you should be, given this is a chair. If you want a wood-only solution you want to use a face-grain plug, not a dowel.

When you use a dowel it is the reverse of the grain direction of the surrounding wood, so you get end grain facing outwards. A screw won't hold in end grain as strongly as into long-grain wood. Now normally this isn't as much of an issue as often thought (here's why¹), but a chair is one of the cases where you want to go for the strongest solution possible, as you've specifically asked for.

In addition there's a practical issue in terms of actually making the new hole.

When drilling like this, where the new drilled hole partially overlaps the previous one and you have a dowel plugging said hole the drill bit has a nasty tendency to wander into the dowel. This is because end grain is much easier to drill into than face grain. So it's hard to ensure the hole ends up quite where you want it, or at right angles to the surface. DAMHIK.

So for me, it's face-grain plug or epoxy for this.

Two disadvantages of going with a face-grain plug however. The first is you need a piece of wood to cut it from², and the second is you have to buy a plug cutter (which usually come in sets). Here's what one of the better types looks like:

^{Veritas tapered-plug cutter.}

Also my concern is redrilling and putting a screw into epoxy. Would it be too brittle? How well would it hold?

Filled epoxy is used extensively in some woodworking contexts, including many applications in modern boatbuilding. On a more small-scale level I've made structural repairs using DIY epoxy wood filler, i.e. wood dust mixed into epoxy. If everything goes well these are extremely durable and can certainly be sound enough to drill into (and tap if necessary) and to hold a screw³.

I'm not certain I'd want to use this here however, but I'd want to see the position of the hole and get a better idea of what it needs to do.

Edit
The last point above led me to thinking.... and unfortunately I realised that any hand-wringing may be largely unnecessary here. This could be a clearance hole. And if so the screw actually only needs to pass through this piece and not hold in it at all. See previous Answer for more on pilot and clearance holes.

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¹ Endgrain screw withdrawal force

² There may be no need to source a separate piece of wood, an unseen surface in any of the chair (or matching table?) members could be fine here.

³ I repaired a ripped-out hole in a kitchen cabinet 2+ years ago and checking it recently there's no sign the Euro-style hinge (on one of the most-used doors) is less secure than any of the others.