At the mirror Tolkien gives Galadriel away as a ring-bearer, why just her do you suppose? Perhaps so that she can best prepare Frodo for the last leg?
My favorite part of our rest here is the gift scene at the end. I so look forward to that part in the extended DVD! From the pictures we have seen already (which I will post later) it looks beautiful.
Pertinent Chapters - Book II
Chapter 6 - Lothlórien
Chapter 7 The Mirror of Galadriel
Chapter 8 - Farewell to Lórien
I think it reflects Frodo's growth as the Ring-bearer. When he was in Rivendell, he was still too inexperienced to perceive Elrond's ring; by the time he gets to Lothlorien he has grown wiser. Galadriel herself comments on this, after she shows him the Mirror: "Yet even so, as Ring-bearer and as one that has borne it on finger and seen that which is hidden, your sight is grown keener. You have perceived my thought more clearly than many that are accounted wise. You saw the Eye of him that holds the Seven and the Nine. And did you not see and recognize the ring upon my finger?"