[Mithadan]

While it is possible that Galadriel may have "offered" Frodo the Ring (which would have made his offer to yield it to her even more striking as a reaction), my guess, and it is nothing more than speculation, is that Frodo's greatest desire was to settle down again with Bilbo. To do this, Frodo would need to be free of the Ring. The easiest route to his being free of the Ring? Give it to Galadriel. So Frodo's offer back to Galadriel, made appropriately before her Mirror, was a reflection of what she suggested to him and what she, herself, desired.