In a book that's so focused on time, Passepartout's watch is a definite talking point. Verne takes time out to discuss and illustrate Passepartout's watch and how important it is to him.
From the very first chapter, Passepartout is always getting the "What time is it?" question. Fogg asks his servant about the time a few minutes after hiring him, and Passepartout pulls a giant silver pocket watch out in order to answer him. Fogg tells him in his usual condescending fashion that the watch is "late." From then on, the watch becomes a central symbol for us, though we quite don't get why until we near the very end.
Detective Fix is also keen to point out that Passepartout's watch is slow. Passepartout is almost insulted: "My watch? A family watch monsieur, which has come down from my great-grandfather! It doesn't vary five minutes in the year, if it's a perfect chronometer" (8.6). How dare Fix speak ill of Passepartout's family heirloom.
Later on, we find out that Passepartout isn't just proud of his pocket watch, he's kind of weird about time altogether. He doesn't change the watch when he crosses into different time zones, as he remarks in a convo with Fix: "'I see how it is, said Fix. 'You have kept London time, which is two hours behind that of Suez. You ought to regulate your watch at noon in each country'" (8.6). Passepartout may circle the globe, but through his watch, he maintains a constant allegiance to his home.
Sir Frances Cromarty also takes the trouble to point out to Passepartout that anyone with sense would change his watch to keep local time:
Sir Francis Cromarty asked Passepartout what time it was; to which, on consulting his watch, he replied that it was three in the morning. This famous timepiece, always regulated on the Greenwich meridian, which was now some seventy-seven degrees westward, was at least four hours slow. Sir Francis corrected Passepartout's time, whereupon the latter made the same remark that he had done to Fix; and upon the general insisting that the watch should be regulated in each new meridian, since he was constantly going eastward […] Passepartout obstinately refused to alter his watch, which he kept at London time. It was an innocent delusion which could harm no one. (9.12)
Okay, so everyone's bugging Passepartout to update the time on his watch, and Passepartout's just not having it. One effect this has on the book is that it keeps time in the mix—and for a book that's racing against time, it's a helpful reminder that the clock is always ticking. But there's something more going on with this watch, too: For all the value placed on time and precision in this book, Passepartout's watch comes to show imprecision. Check out what happens at the end of the story:
Phileas Fogg had, without suspecting it, gained one day on his journey, and this merely because he had travelled constantly eastward; he would on the contrary, have lost a day, had he gone in the opposite direction, that is, westward.
And Passepartout's famous family watch, which had always kept London time, would have betrayed this fact, if it had marked the days as well as the hours and minutes. (37.6-8)
Passepartout is always talking about his watch, checking the time and defending his decision to keep his timepiece oriented toward home. And yet as soon as they journey outside their starting time zone, Passepartout's watch grows increasingly inaccurate without anyone in the traveling party fully realizing this fact. For all their effort to operate on schedule, then, Fogg's victory ultimately relies on luck. Ha.