Why was Boromir left out in receiving gifts from Galadriel?

The short answer is that each of the gifts that were filmed played significant parts in the future movies whereas Boromir's gift (in the book it was a "belt of gold") had no future importance.

In the Extended Director's Commentary, Peter Jackson spoke to the reasons why each scene was originally filmed (as well as his disappointment that they didn't make the theatrical cut and his pleasure that they made the DVD):

PJ : The cloaks that the Fellowship wear, erm, were given to them in Lothlorien and in the movie version they're just suddenly are wearing these cloaks and there's no explanation of how they got them. The Lembas bread is significant because, ah, Lembas bread returns to the story in the second and third film and we filmed sequences in those movies with the lembas bread and yet we never introduced it in the theatrical version of the Fellowship of the Ring so this is another reason why I'm glad that this DVD has an the opportunity of being seen by people prior to the release of the Two Towers 'cause actually one of the first scenes in the Two Towers, between Frodo and Sam, um, involves them eating Lembas Bread so at least people will have some concept of what that actually is now.

This is where Aragorn gets his dagger, which he uses throughout the rest of this movie and the next two films so it's nice to be able to see it being a gift from Celeborn.

Legolas gets his bow. Now Merry and Pippin get these daggers which again form a significant moment in the Two Towers so I'm glad that people are able see these daggers being given. All of the gifts that Galadriel gives the Fellowship are actually used in the next two movies in different ways and so I was pretty adamant that I wanted to include all of these scenes in the DVD because this rope, for instance that Sam has been given does play a part in the Two Towers and, uh, y'know, the hope is that most people will get to look at this DVD and understand where these things come from, as they look at the next films

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