The first mention of a special time piece in the Weasley home is from Chamber of Secrets, page 34. However, this appears to be a different clock.
The clock on the wall opposite him had only one hand and no numbers at all. Written around the edge were things like Time to make tea, Time to feed the chickens, and You're late.
The first specific description of the family clock is from Goblet of Fire (pages. 151-153)
Mrs. Weasley glanced at the grandfather clock in the corner. Harry liked this clock. It was completely useless if you wanted to know the time, but otherwise very informative. It had nine golden hands, and each of them was engraved with one of the Weasley family’s names. There were no numerals around the face, but descriptions of where each family member might be. “Home,” “school,” and “work” were there, but there was also “traveling,” “lost,” “hospital,” “prison,” and, in the position where the number twelve would be on a normal clock, “mortal peril.”
Eight of the hands were currently pointing to the “home” position, but Mr. Weasley’s, which was the longest, was still pointing to “work.”
. . .
“Oh your father’s coming!” she said suddenly, looking up at the clock again.
Mr. Weasley’s hand had suddenly spun from “work” to “traveling”; a second later it had shuddered to a halt on “home” with the others, and they heard him calling from the kitchen.
It appears again in Order of the Phoenix (pages 471-472)
”And Dumbledore – what about Molly?” said Professor McGonagall, pausing at the door.
“That will be a job for Fawkes when he has finished keeping a lookout for anybody approaching,” said Dumbledore. “But she may already know . . . that excellent clock of hers . . .”
Harry knew Dumbledore was referring to the clock that told, not the time, but the whereabouts and conditions of the various Weasley family members, and with a pang he thought that Mr. Weasley’s hand must, even now, be pointing at “mortal peril.”
Finally it's described in Half-Blood Prince (pages. 85-88) This last description seems to conflict with the more detailed description in Goblet of Fire. In GofF, the clock is a Grandfather clock. In HBP, it is small enough to carry around. The GofF description seemed geared at giving ambiance to the house. The HBP description seemed more intertwined with plot and character. I chose to follow this one more closely and finally settled on a mantel style clock for my own design.
She turned to look at a large clock that was perched awkwardly on top of a pile of sheets in the washing basket at the end of the table. Harry recognized it at once: It had nine hands, each inscribed with the name of a family member, and usually hung on the Weasley’s sitting room wall, though its current position suggested that Mrs. Weasley had taken to carrying it around the house with her. Every single one of the nine hands was now pointing at “mortal peril.”
“It’s been like that for a while now,” said Mrs. Weasley, in an unconvincingly casual voice, “ever since You-Know-Who came back into the open. I suppose everybody’s in mortal danger now . . . . I don’t think it can be just our family . . . but I don’t know anyone else who’s got a clock like this, so I can’t check. Oh!”
With a sudden exclamation she pointed at the clock’s face. Mr. Weasley’s hand had switched to “traveling.”
. . .
Harry saw Mrs. Weasley glance at the clock in the washing basket as they left the kitchen. All the hands were once again at “mortal peril.”