The huge, chronograph-like watch that Esmé sports is really the only continuing visual symbol that we encounter in the story. From the beginning, the oddness of the huge watch on the girl's tiny wrist disturbs the narrator, and unsettles us, his readers, as well. When Esmé explains that it belonged to her deceased father, who gave it to her "purely as a momento" (76), it all clicks into place; the watch, which Esmé rather self-consciously but proudly wears, is the visually disturbing representation of her hidden (and not-so-hidden) and unresolved grief for her father. By carrying around this unwieldy object, she maintains some connection with him, despite the fact that she won't admit that she actually misses him, instead emphasizing the fact that Charles, the younger sibling, misses their parents a great deal.
Why, then, does Esmé send the watch to Sergeant X? One very literal interpretation might be that she sees in him another father figure, and therefore associates him with the watch. However, we think it's more subtle than that – Esmé sends it as a good luck token, and invests the watch with all of her complicated emotions about the war (her love and longing for her father, her fears and hopes about the outcome of the war, and her concern for Sergeant X).