In Peter Jackson's Fellowship of the Ring, as the fellowship is leaving Lothlorien, Galadriel gives gifts to every member -- the Elven cloaks; the Elven bow; the knives; the Elven rope; etc -- except for Gimli. Gimli asks for one of Galadriel's hairs as a gift.
Why did Gimli want Galadriel's hair? Would it have served a utilitarian purpose?
I don't know what the movie showed, but this is very clear in the book. This is Gimli's first reaction to Galadriel (book 2, chapter 7):
She looked upon Gimli, who sat glowering and sad, and she smiled. And the Dwarf, hearing the names given in his own ancient tongue, looked up and met her eyes; and it seemed to him that he looked suddenly into the heart of an enemy and saw there love and understanding. Wonder came into his face, and then he smiled in answer.
Gimli himself explains why he wants a strand of hair:
‘Treasure it, Lady,’ he answered, ‘in memory of your words to me at our first meeting. And if ever I return to the smithies of my home, it shall be set in imperishable crystal to be an heirloom of my house, and a pledge of good will between the Mountain and the Wood until the end of days.’
You'll see more of that in book 3 chapter 2, when Gimli reacts to Éomer's less-than-gushing description of Galadriel:
You speak evil of that which is fair beyond the reach of your thought, and only little wit can excuse you.
In short, Gimli is smitten.
Gimli thought that Galadriel was the most beautiful thing he has seen so far.
Gimli's opinion of Elves drastically changed when he met Galadriel, co-ruler of Lothlórien: her beauty, kindness, and understanding impressed Gimli so much that, when given the opportunity to ask for whatever he wished, he responded that being able to see her and hear her gentle words was a gift enough.
...he admits that he desires a single strand of her golden hair, so that he might treasure it and preserve it as an heirloom of his house.
Tolkien made it very clear in his story that Gimli was enchanted (not literally) by Galadriel. He thought she was the fairest creature in Middle Earth. He sent for his axe when Faramir disagreed, but relented when he learned that Arwen was the competition. There was no utilitarian purpose. It was a token of her that he wished to have as a remembrance and a way to honor her.
Galadriel was one of the two most powerful of all the Noldor - equalled only by her cousin Feanor, whom she hated. Her inner magical and athletic stature and strength were matched only by her outward beauty. As part Vanyar she was also notably golden haired - something almost totally unknown in other elves; Legolas was for instance actually black-haired. Gimli, as a Dwarf and a smith and therefore one who innately valued material beauty beyond anything else was struck by all of this and he wasn't the first.
The light of Laurelin and Telperion falling on Galadriel's hair was what inspired Feanor to make the Silmarils in the first place. He actually begged her for a lock of hair just as Gimli would however many hundreds of thousands if not millions of years later. Galadriel turned Feanor down however.
