At his workshop in rural County Mayo, Neville O’Farrell has been working on a display box of walnut with a bog oak veneer for a special timepiece.

He operates Neville O’Farrell Designs, a business that he and his wife, Trish, founded in 2010. He creates handmade boxes from native and exotic hardwoods, with prices starting at 1,800 euros ($2,020); Ms. O’Farrell handles the finishing work and business details.

Most of their customers are in the United States and the Middle East. “New Yorkers and Californians order jewelry and watch boxes,” Mr. O’Farrell said. “Texans order humidors and boxes for their hunting guns.” And, he added, Saudis commission ornate humidors.

The walnut box is for Mr. O’Farrell’s only Irish client: Stephen McGonigle, a watchmaker and the owner of McGonigle Watches, based in Switzerland.

Mr. McGonigle commissioned it in May for a Ceol Minute Repeater watch that he is making for a collector in San Francisco (starting at 280,000 Swiss francs, or $326,155, excluding taxes). Ceol — the Irish word for music — refers to the watch’s repeater, a mechanism that chimes the hours, quarters and minutes on demand.

The collector does not have Irish roots, but he liked the Celtic decoration typical of Mr. McGonigle’s watches and chose an abstract pattern of birds that the watchmaker is engraving on the timepiece’s dial and its bridges, the term for the plates that hold the inner workings and are visible through the case back.

The pattern was designed by Frances McGonigle, an artist and the watchmaker’s eldest sister, who was inspired by the art that medieval monks created for the books of Kells and Durrow. “The old manuscripts are full of mythical birds, and birdsong speaks to the ‘Ceol’ of the watch,” she said. “I love how the bridges in the watch mimic the long beak of a bird.”

The client wanted the box, which is 111 millimeters high by 350 millimeters wide and 250 millimeters deep (roughly 4.5 inches by 14 inches by 10 inches), to be made of bog oak, a dark wood salvaged from Irish peat bogs after thousands of years. But Mr. O’Farrell, 56, said that bog oak came “in lumps” and would be unstable. He used walnut and a bog oak veneer instead.

Ciaran McGill, a craftsman from the Donegal specialty shop The Veneerist, executed the marquetry decorations, using bog oak and a pale piece of figured sycamore, which is often used as a veneer for string instruments. “It’s a bit like a jigsaw puzzle,” he said.

It took two days for him to inlay the McGonigle logo on the lid and to add the bird pattern to the lid and sides. Inside, he spelled out McGonigle on the left rim and Ireland on the right in the Ogham alphabet, which was used to write the earliest form of Irish, dating from the fourth century.

Mr. O’Farrell said he hoped to complete the box by the end of this month; most take from six to eight weeks, depending on size.

Achieving a high gloss finish to a box’s polyester glaze is hardest part, he said. Ms. O’Farrell sands for two days and then, using a gritty compound on a cotton rag, polishes for 90 minutes, repeating the process 20 times.

It can go terribly wrong. “If one tiny speck of dust contaminates the rag,” Mr. O’Farrell said, “it will scratch the wood.” The box would then have to be stripped and the process repeated. “That’s when you hear screeching and cursing!” he said with a laugh.