Upon leaving the lands of Lothlorien, Lady Galadriel gives each of the Fellowship a gift and each gift is given with the receiver in mind and not at the receiver's request. But when she comes to Gimli, Lady Galadirel instead asks him what he would want.
The fact that Lady Galadriel must ask Gimli what he would like indicates that she does not have a clue as to what to give him. This is understandable, given the animosity between the two races.
When asked once, Gimli eloquently declines any gift. He states that just seeing the Lady Galadriel and hearing her words is enough of a gift. These words, of course, flatter Lady Galadriel and she returns an as eloquent compliment.
But she does not let Gimli off so easy. She insists that he choose a gift.
For Gimli, this is a tricky spot. He has fallen platonically in love with Lady Galadriel and wishes only to be in her favor. To request some physical object, such as the typical jewels, weapons or magical items, may cause him to look like the greedy Dwarf that many standing around her believe him to be.
But, on the other hand, he can not simply brush her request aside. He has already tried it once and she has insisted. To not fulfill her request, he risks bringing on her disfavor.
So Gimli requests what should be a rather mundane and simple item. A single strand of her hair. Who could argue with that?
Except Galadriel's hair has been requested as a token of love before, from a high ranking Feanor, and she declined the request from one of her own kind. And yet, she gives Gimli, not just one strand, but three!
There are several reasons why she may have done this.
First of all, she did ask Gimli what he wanted. She would look just a little silly reneging on her own insisted request that he name what he wanted.
Second, Gimli is just so humble when he asks for it. He says, "I do not ask for such a gift. But you commanded me to name my desire." He makes sure that she knows that he wants the hair but he knows that he is not worthy. He is not so bold as to ask, but is simply answering her question.
Third, Gimli is just so darn flattering about it. He says that her hair "surpasses the gold of the earth as the stars surpass the gems of the mine".
He also tells her that he will encase the hair in "imperishable crystal" to be handed down as a high treasure from generation to generation. What woman can argue with being remembered forever? Even High Elves are not impervious to flattery.
All we know is that Lady Galadriel is so taken with Gimli's request and flattery, that she grants him not one hair, but three.