Galadriel's intentions.

[Boromir88]

That's very possible Dol Guldur. Gandalf says in The White Rider "She told me he (Boromir) was in peril." Perhaps Galadriel caught hints of the Fellowship beginning to fall apart. She tested each member to see who would stick with Frodo, and started the "breaking of the Fellowship." Not for evil purposes, but possibly to realize the sooner the better? Perhaps she just awoke in Boromir the "sleeping giant" because maybe the longer it went "dormant" the worse it would be?

Edit: The fellowship was also hit with constant bad luck, and it almost seemed destined to break apart. I mean first the wolf attack, then the storms on Caradhras, then all the evil in Moria, then the problems down Anduin and the argument between Aragorn and Boromir. It just seemed constant doom hung around the Fellowship and Galadriel realized it was best to break it.

For Aragorn, it's not that he might have not been loyal to Frodo (I mean if he wanted to he could have easily taken the ring in Bree and going to Rivendell), but just that his path was ment to be different. Not to go along with Frodo, but to save Minas Tirith. As Aragorn makes an oath to Boromir....

"No!" said Aragorn, taking his hand and kissing his brow. "You have conquered. Few have gained such victory. Be at peace! Minas Tirith shall not fall."~emphasis mine

With that oath, I think it becomes clear that he wasn't ment to stick with Frodo, and follow him to the end, but to try and save Gondor.

[Keeper of Dol Guldur]

I wonder then, if Galadriel realized that it would take Boromir's death for Aragorn to resolve to go to Minas Tirith, now that Gandalf had fallen.

Because Aragorn had always intended to return to Minas Tirith with Boromir ... but the second Gandalf was lost and he was forced to become guide to Frodo, he sort of got lost in the task at hand, and obviously made many comments about not knowing which road to take.

I am not Gandalf, and though I have tried to bear his part, I do not know what design or hope he had for his hour, if indeed he had any.

Aragorn then suggested that he and Gimli accompany Frodo and Sam, while the others went to Minas Tirith. But he only suggested this out of loyalty to Frodo.

What good would Anduril and the returning King be in Mordor? Or what use would Gandalf the Grey have been, for that matter? Gandalf couldn't have concealed his power from the malice of Sauron, not once he entered Sauron's lands.

[Lhunardawen]

Boromir88--

From what you've said I realized that if anyone, other than Frodo, was greatly helped by what Galadriel did, it was Aragorn. After all, he was the one most torn, so to speak. Frodo and Boromir knew what each of them wanted to do. Sam was certain where his loyalty lies. The rest were not yet decided, but like as not they would follow Aragorn, wherever he went. And Aragorn has been burdened with the leadership of the Company ever since Gandalf fell in Moria. He was loyal to Frodo and the Quest, but as the heir to the throne of Gondor he also desired to go to Minas Tirith and "save his people". Because of the sequence of events that led to Boromir's actions and, eventually, his death, Aragorn has decided where his road really leads him. And as Keeper of Dol Guldur said:

What good would Anduril and the returning King be in Mordor?

no doubt he has chosen the best path he could possibly choose. All things indeed work together for good.

So, Galadriel loves her (soon-to-be) grandson-in-law that much, huh?

[Boromir88]

But would you say that if it wasn't for Boromir trying to take the Ring from Frodo, and causing Frodo wanting to get away, Aragorn still would have wanted to go to Minas Tirith? Do you think the breaking of the Fellowship, and the events, is what led Aragorn knowing what his path now was? Or was it just fate, that even if the Fellowship didn't break apart, something down the road would lead Aragorn to his path of becoming King?