My wife's mother has some fine dining chairs, probably around 50 years of age. The wood is some type of mahogany.

Anyway the inevitable has happened and the joints between the rear legs and the rails are busted. There are 3 dowels on each side, around 8mm in diameter, reinforced with a triangular block. IMO, the design is quite weak as there is no stretcher so the joint tends to rack, and bingo, all of the dowels have broken.

Now I could attempt to drill the dowels out and redo them, but where is the fun in that, and knowing my luck the dowel holes will not line up. So I'm thinking a Domino joint would be superior as there will be more material and a greater glue surface.

The legs and the rails are around 21mm thick, the rails are around 50mm high from the bottom to where some fancy profiling starts.

Using the rule of one-third thickness, a domino around 7mm would be ideal, but does not exist, so I'm likely to use an 8mm.
However the width of that domino is only 22mm, I think a wider one (up to around 30mm) would provide a stronger repair.
An alternative would be to use a 10mm domino, which has a width of 24mm, but risks splitting the leg or rails as the wall thickness would be less than 6mm.
The third option would be to use the 8mm cutter to make a mortise around 30mm wide and then make my own domino/spline to fit.

Any feedback and advice would be good, as I'd like to get this job done next weekend in time for festive gatherings at the end of the year.

I would dig out as much of the old dowels as possible. If the original glue was hide glue, hot water will weaken it and you can remove the remaining debris. I would very carefully drill out the holes and trim fit (no glue) replacement dowels.

If you have a decent fit, you can just glue in new dowels, clamp, and let dry.

If the dowels are wobbly (holes are too big), I would try drilling them out until I got a decent fit on a larger dowel.

Using Dominos looks (from the photo) difficult in this case because you lack much of reference surfaces.

Thanks for the advice.  I'll carefully remove the dowels and have a go at replacing them. 
It looks like the key points are removing the old ones, cleaning out glue residues, and ensuring a good fit.
I do not have access to hide glue but do have Titebond 3, Epoxy and Resorcinol.

Two part epoxy seem to work better than TiteBond if there are small gaps.  If the dowel fit is snug, Titebond will be ok.

I’ve had good luck using surgical tubing for clamping odd shaped things. It is really stretchy and can exert a lot of force. It’s used for serious slingshots.

I have strengthened chair joints by adding pocket hole screws on the bottom or inside of pieces. Use glue along with the screws. Also need to make sure the part is wide enough to accept the screw without creating other damage.

Seth

Thanks for all the helpful replies. I will repair by redoing the dowels, hopefully with a stronger wood.
The joints are reinforced with triangular blocks, which are glued and screwed. The glue had broken there too.
That glue joint was never going to be very strong as the block is at 45 degrees to the grain, but there is a reasonable surface area.  (Edited: Although looking at the failure, it appears the wood failed here rather than the glue.)

The chairs have been "repaired" in the past, although not by me. I'm looking to do a durable job.

Use a drilling jig and visually align the bushing with the dowel. Re-drill and glue up using dowels.  If the original lasted 50 years, then the construction method is pretty well proven. 

I have a couple of dowel jigs.  I would probably used the one with the clear base.  Available from Amazon.com for $20.00.  It is not always my go-to drill guided (it has pieces that can get lost), but for visual alignment, it is the bees knees.

The drilling jig I showed above has a wide base.  You should be able to shim it up to get the correct angle.

The big problem with that jig is that it has “pieces”, and I frequently lose “pieces”.  I keep it in the original box and take it out when I need it.

Mostly I prefer the Milescraft drilling jig ($8.00 at Lowes).  The Kreg version, a near clone, is just $7.00 at Lowes.  No pieces to lose.  And they appear to be accurate. 

Wolfcraft also makes a low cost clone.  None have the tactile appeal of the Big Gator jigs, but they appear to be just as accurate.

It's clear from this second pic that the fixed joint didn't have a good bonding, likely due to inadequate clamping and even insufficient glue used. Unlike the brace (which you could see the wood fibre on the glue), the white glue on the dowel part is pretty intact.

Yes I found the glue on the rails (shown) had bonded well so i'm going to redrill the rails. The dowels in the legs could be prised out.