Hello all!
I am sure that this particular subject has been discussed many a time, but I wanted to go ahead to propose my interpretation of the aforementioned item.
Throughout Utena, Miki is seen using a stopwatch, which he stops at various times. On first inspection, the stopwatch seems to imply two, surface-level items:

1)    Miki holds the role of Student Council Secretary/Minuteman, and
2)    Audience members are only presented with a small slice of the Student Council’s conversations.

I believe, however, that the stopwatch has far greater significance than it initially suggests – in fact, I believe that it provides another, telling glance into Miki’s character, namely, the issues with which he struggles.

The reason that Miki duels is, invariably, derived from his desire to [re]capture past emotions, experiences, and entities, all of which remained firmly placed within his nostalgic, but ignorant, mind. Although Miki’s “shining thing,"  i.e., his time in the Sunlit Garden with Kozue, does exist, it only resides within the past and his brain, neither of which have a [direct] impact upon the present state of affairs; in a manner of speaking, this world, and the entities and events which inhabited it, no longer exist.

Now, before I continue, let me clarify what I mean by “existing."  An entity, event, or item “exists"  if it is a resident of the present, a definition that is derived from a Presentist perspective of time. Such a view is, invariably, subject to debate, but it is one that is inherently useful in this discussion.

Now, back to Miki. Throughout his interactions with Anthy, including his duels, Miki is trying [and failing] to resurrect past emotions and experiences, none of which, invariably, exist; in other words, he is striving for the impossible, which is signified by the color of his rose.

Now how, may I ask, does this relate to his stopwatch? Well, one measures time using timepieces, which aid in the creation of historical [and menial] records. However, like Miki’s memories, the events and entities that are referenced by these records are, in a manner of speaking, nonexistent – that is, they no longer exist within the present, but remain relics of the past. We may possess records [and memories] of these items, but it is impossible to [re]create them in their original states; similarly, Miki cannot, in any way, resurrect the Sunlit Garden, despite all of his efforts to do so. All that Miki strives for, and does, is, invariably, in vain, and only by moving from a nostalgic, past-oriented mindset to an open-minded, present-oriented mindset can he hope to rebuild his relationship with his sister, Kozue.

Perhaps Miki chose this position, i.e., Secretary, because of his obsession with time, i.e., the past?