The fellowship comes to Lothlorien at a time of desperate need, when they are on the brink of breaking. They have just lost Gandalf to the Balrog in Moria, and they don’t know which direction he meant for them to take, so they have never been more in need of Galadriel’s guidance. Not only does she welcome them into her lands, give them food and places to sleep, and time to rest and heal both their physical and emotional wounds, she also gives each member of the company a parting gift.
These gifts are not trivial tokens, they are carefully thought out and generously bestowed treasures and each one has an essential purpose in the fellowship's journey, whether it is the lembas bread that keeps Frodo and Sam alive during their dangerous quest to Mount Doom to destroy the ring of power, or the ring of Arwen that she gives to Aragorn to remind him that he can overcome the greatest of challenges. But of all the gifts she gives, there is one that is very different from the others, and also has a very different symbolism.
Most of the things that she shares are objects, including the elven cloaks to keep the company safe during adverse weather and help them stay disguised from the enemy, and a whole hoard of weapons, including daggers, bows, and arrows, and even a golden belt for Boromir of Gondor, as well as the light of the most beloved star, that saves both Frodo and Sam from imminent death. But as she passes along the line of people before her, she stops in front of Gimli with nothing to offer him.
It has been hundreds of years since a dwarf has set foot in her lands, and there has always been an uneasy rivalry between the two races, with much history and bad blood behind it. So when she gives him the chance to pick a gift, anything that his heart desires, this is a true honor indeed. And perhaps, thanks to her powers of foresight, she already knows what he will ask, and it is because he asks for something so modest, so sweet, that she is happy to bestow it.
But the three golden hairs that Galadriel gifts Gimli at his behest have far more significance than their just being very pure and innocent requests, rather than all the gold or power or wisdom he could have asked for. Any fans who have read the Silmarillion will know that Feanor, one of the greatest elves, and most amazing crafters of the world had made a similar request of the golden lady thousands of years earlier.
Feanor was so inspired by Galadriel’s grace and beauty, that he wished to use strands of her hair in the making of his three Silmarils, the most precious gems ever created in Middle Earth. He begged her three times for a single strand of her hair, but even though she was young and inexperienced at this point, she looked into his soul and saw that his intentions were not fair and benevolent, but were driven by greed and a desire to possess something more precious than the other elves of the time. She, therefore, refused his request, causing much wrath and anguish, and he was forced to find other precious substances to infuse into his gems.
It is incredible then, that Galadriel would give this gift to Gimli. Her husband Celeborn is the only other being in all of her many long years who has been able to so much as touch the golden strands of hair on her head, let alone pluck them out and keep them.
So when she allows this incredible rarity to Gimli, she essentially gives him the three hairs that Feanor demanded of her, because she has looked into his soul and found that he has a good heart and that he desires them for nothing more than to treasure the beautiful days he spent in her lands. Here, she finds a dwarf who is worthy of ending the hundreds of years-long rivalry between their races, who proves that a dwarf can be loyal and brave, and all of the amazing qualities that Gimli shows, despite so many of his people simply being greedy and power-hungry.
It is also thought that many of the elves of Middle Earth know about Galadriel’s altercation with Feanor, and that at this moment, Legolas sees what an incredible honor the she-elf has given his companion, and realizes that he can finally trust him and put aside their differences. This is the start of their life-long friendship, and a strengthening of the fellowship that will survive many dangerous times ahead together, all thanks to Galadriel’s three golden hairs.