[Boromir88]

morm, excellent as usual. Now what Galadriel offered to Boromir we can't be sure about, because we aren't told. And certainly after Lorien a change went through Boromir. Prior to this, beside what happens at the Council Boromir had shown no pull towards the Ring. However, after Lorien we can tell it's affecting him, he's showing signs of nervousness (biting his nails) and he's questioning Frodo.

"Boromir, O Boromir!" he cried. "What did she say to you, the Lady that dies not? What did she see? What woke in your heart then? Why went you eve to Laurelindorenan, and came not by your own road, upon the horses of Rohan riding home in the morning?"~The Window on the West

So, something happens to Boromir after Lorien, and I'm thinking it has to do with Galadriel, and Galadriel may actually have offered Boromir the Ring.

We later find out that Galadriel has also desired the Ring for a long time. And what's interesting is Boromir calls it "a test." Tolkien has referred to "test/trial" as when one has an oppurtunity of taking the Ring, do they are don't they. Galadriel "passed the test."

It did not seem possible to Faramir that any one in Godnor could rival Boromir, heir of Denethor, Captain of the White Tower; and of like mind was Boromir. Yet it proved otherwise at the test.

Galadriel, in my opinion, offered the ring to Boromir but his promise to be true to his word was true in a sense that he would stick to the quest, that is what he gave his word on; however he knew that by staying on this quest an opportunity may present itself where he could take possesion of the ring.~mormegil

The thing is is that Boromir did not give his word about staying with the quest. Remember, no one was bound to sticking with Frodo. Both Boromir and Aragorn were in the Fellowship because they were heading to Minas Tirith. But, what does make me think that Galadriel offered Boromir the Ring is as you point out "staying true to his word"...Not in the sense of staying with the Fellowship, but in the sense of protecting Frodo. There was no word keeping members of the Fellowship together, but while in the Fellowship there was the expectancy that the members were to do whatever they can to help Frodo. And of course that word would be broken (and he eventually does break it) if Boromir tried to take the Ring from Frodo.