Galadriel's intentions.

[Hot, crispy nice hobbit]

I am not sure if I am correct to put it this way, but I think that Galadriel had already explained her intentions to Frodo...

Frodo bent his head.'And what do you wish?' he said at last.
'That what should shall be,' she answered.

Apparently by that, Galadriel seem to indicate that she was only an instrument of fate: whether to test the Fellowship of their faith, or to assist them. Somewhere along that line, she was also seeking to take her final exams... It is interesting to note that after Frodo 'tested' her, she said that:

'Let us return!' she said. 'In the morning you must depart, for now we have chosen, and the tides of fate are flowing.'

Perhaps one of the choices that Galadriel had laid out before Frodo had been to surrender the Ring to herself. It was assumed, of course, that other members of the Fellowship had already made their choices, so the only thing that was still left to settle was Galadriel's own desire 'to rule Middle-Earth'. She herself understood that her own decision would probably be the most critical of the series of 'tests' that she made the others go through. So in end, she passed her exams...