I agree about the Legolas-Gimli relationship being depicted well.
Also, the scene with the hobbits in the bar at the end of ROTK- that is such a well done scene in my opinion! It shows the hobbits a bit apart from the rest of the people in the bar, which by contrast, underlines their friendship with each other. Only they know what they went through, and this knowledge brings them together. But at the same time, they're still the kind of friends they were before they set out, ready to rejoice in drinking an ale (compare with scene in FOTR in the bar).
Also another friendship that stood out for me is (don't laugh) Boromir and Aragorn. This mostly works because of Aragorn's brilliant skills in dealing with people. His modesty as a character in the movie really pays off when it comes to this relationship. Meaning, Aragorn from the book and Boromir from the book didn't get along so well together, they were always a bit cold and at a constant disagreement. Of course, Boromir from the movie was far more good natured that Boromir in the book.
There are scenes like the one in Lothorien, when they talk together, or when they just sit together, talking and smoking, in Moria, waiting for Gandalf that tells you they're good friends; so it doesn't come as far fetched to see Aragorn cry when Boromir dies.