I don't believe it was Eru who sent Gandalf back, for two reasons. One, Manwe or another Vala would be perfectly capable of doing so, and as they were the ones who sent Gandalf over in the first place, it seems most likely they’d be the ones to send him back. Two, that would involve Eru directly interferring, for lack of a better word, in Arda. It's pretty debated as to whether he would do this or not, but the general idea is that he wouldn't, at least for something like this. (see the Athrabeth for more details there)
Isn't it possible that Galadriel realized that Gandalf might be back, and sent an Eagle to keep an eye out for him? I believe Galadriel does say that she "cannot see [Gandalf] from afar… a grey mist is about him, and the ways of his feet and his mind are hidden from [her]." So I'm pretty sure she can't just use the Mirror to find him, nor the Ring. An Eagle would be the best way to look for him.
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