Historyedit edit source. Before the Company of the Ring left
The short answer is that each of the gifts that were in the film played significant parts in the future movies whereas Boromir's gift had no future importance. In the book it was a "belt of gold".
Galadriel then presents the Fellowship with additional gifts. To Aragorn she gives a sheath for his sword, Andúril, and a green gem in a silver brooch. Boromir, Merry, and Pippin each receive belts of silver or gold, while Legolas receives a longer, stouter bow.
She didn't tell him anything. She just tested him by offering him what he desired most, which for Boromir was the One Ring; but she didn't know what it was. However while he refused it (as he said, since as Sam said, “lying wasn't his way”), he still wanted the Ring.
Also, given how Boromir is (and will be) tempted by the (gold) ring, which Galadriel plays a part in, it's interesting how her gift to him is a gold belt, which is almost like a ring...
Boromir was put under great stress by his father to retrieve the Ring, and he let the Ring take control of his mind. Because of this, he is ashamed and disappointed in himself. This proves the Boromir is at his core a good person. Boromir knows what he has been thinking is wrong, and so he cries.
Though the stark contrast between good and evil are clear in much of The Lord of the Rings, the lines between them blur in the character of Boromir. He was a good man corrupted by the evil of the Ring. His conflicts and flaws highlight not only the Ring's deadly pull but also Boromir's human nature.
It's also possible that Boromir had a touch of elvish blood on the father's side anyway. The House of Hurin is said to be of high Numenorian lineage, and may have intermarried with offspring of Elros' line. But it's so far removed that the elvish touch is beyond real trace.
Like his father, he is "cursed" with Isildur's greed and is obsessed with the Ring. He believes that with the Ring in Gondor's possession, they could win the war against Sauron. He joins the fellowship so he could take the ring from Frodo and return it to Gondor but becomes attached to his comrades.
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.
A Galadhrim bow strung with elf hair and arrows for Legolas.
"Legolas Greenleaf long under tree, In joy thou hast lived, Beware of the Sea! If thou hearest the cry of the gull on the shore, Thy heart shall then rest in the forest no more."
the Evenstar, a jewel which Arwen gives to her love Aragorn; invented for The Lord of the Rings film series directed by Peter Jackson.
That's right: Celeborn is dead. The Rings of Power deviates quite often from J.R.R. Tolkien's canon, but this might be one of the most seismic shifts yet — beyond the literally earthshaking formation of Mordor, of course.
Galadriel Was Ashamed Sauron Tricked Her
The fact that she had saved his life, become his friend, and potentially even developed a little bit of a crush might have just been too shameful for Galadriel to share immediately. There could have also been a darker reason that Galadriel didn't want to confess the deception.
But when it becomes clear to Galadriel that Halbrand/Sauron cannot tell the difference between “saving” and “ruling” Middle-earth, she rejects his proposal. Sauron enchants Galadriel, and when she awakes, he has disappeared.
In fact, Bilbo's possession of the Ring in particular was characterized by pity, as he encountered Gollum and saw what he had become. The compassion he shows to Gollum by not killing him could have been a large factor in why the Ring did not corrupt him the way it did other Ring-bearers.
Because of their intense love of the earth, the Ring does not sense hobbits as a threat, which is why they were able to be overlooked and complete the quest.
Boromir was the weak link within the Fellowship due to his desire to see it as a tool of warfare: glory and pride were its avenues to his heart. Over the months the questions and doubts grow, until the Ring had a gateway to influence him fully.
Why did Aragorn live so long? Aragorn is not half Elf, although he is a descendant of Elros, who is half Elf (and the brother of Elrond, the half-Elf who raised him), which explains why Aragorn's life span is unusually long.
After the loss of Gondor's last King, Eärnur, the Stewards of the House of Húrin took up the rule of the South-kingdom. Denethor, Boromir and Faramir were all descended from this House.
Boromir is the son of Denethor II and Lady Finduilas of Dol Amroth. He had a younger brother Faramir. A year after Faramir was born their father became the ruling Steward of Gondor, and Boromir became heir apparent, inheriting the Horn of Gondor.
That Frodo is unable to destroy the Ring when he reaches the Cracks of Doom is both inevitable and devastating. After all, poor Frodo couldn't even toss the Ring into the fire at Bag End!
Why didn't Aragorn heal Boromir like he did for Faramir in the Houses of Healing? Boromir's wounds are fatal, resulting from many arrows through the chest. There is a limit to Aragorn's ability to heal. Faramir, Éowyn and Merry each had the 'black breath' sickness, which could be treated and cleansed away.
The redditor notes that, after arriving in Rivendell, Boromir picks up the shards of Narsil (the sword that Isildur used to cut the Ring from Sauron's finger) and accidentally cuts his own finger on the exact part of the blade that cut Sauron.