If I was doing this repair I would find a similar piece of wood (looks like mahogany) and use what is called a plug cutter to make a plug for the hole. We use them for removing a circular segment of wood to make a plug that will match the grain of the piece that's being repaired. Yes, a piece of dowel will do the job, but it will be very noticeable. That hole looks like it could be filled with about a 1 1/4" plug, maybe less. If you choose this route, be careful not to buy a tapered plug cutter, they are a pain!
If there are any cabinet makers or joiners in your neighbourhood you could even ask if they will cut a plug for you to fit the hole, in exchange for a six-pack of his or her favourite beverege!

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To make the hole match the plug you will need to use a 1/2" power drill with a 1 1/4" Forstner bit on a low or medium speed so the bit is easier to control. You'll have to drill it out carefully so that the hole stays as round as possible which will make a cleaner repair.
I can't help but recommend Titebond 3 wood glue - this is fantastic wood glue!
It looks like a woodpecker pecked a hole in it!
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