There is a dilemma, a two-fold problem, because one doesn't want to try to speak so "profoundly" as to be vacuous, nor does one want to speak so "simply" as to be trite. One can say too much in saying too little, and say too little in a flood of words. Nor is it best to simply say "what is on one's heart," since grief and loss often cause a myriad of emotions, not all of which ought to receive expression, least of all to those who are closest to the pain. None of us want to be like Job's "friends."
It is a somewhat better approach to think about what you would want to hear if you were closest to the pain. As a Christian, I would want to hear most from God, and so what would be more appropriate than verses that speak of God's faithfulness, the pain experienced by His people, or by Himself, and other words that God has revealed to us of Himself and of ourselves, which we find in the Scriptures. Of course some thought should be put into the selection or selections of Scripture. (If one were consoling an unbeliever, then one's choice of Scripture would need a separate kind of scrutiny.)
I would also want to hear from those who love me. It is a tendency of some to avoid people who are grieving. Sometimes the motive for doing so is obviously selfish; to avoid any of the discomfort of being close to loss and pain. Sometimes the motive is less obviously selfish; to avoid out of fear of doing something to make things worse. A helpful analogy here is to consider a bruised plant. A bruised plant is in a precarious situation. If it is handled too roughly, it may break. If it is handled not at all, it may wither, or break from lack of support, or both. A bruised plant needs attention, it needs care, but it requires a wise hand. The point is that the more love one has for the plant, the more necessary it is to attend to it, since love is itself a governor of the type of care that acts wisely. It must also be said that such love isn't an emotion (though emotions may attend it), but a manner of relationship with consequent volitions. Moreover, the plant needs most of all the care of the one who has provided the most love already, that is, the one who has poured out his effort to cultivate the plant. The proportion to which you have loved the persons closest to the pain of loss and grief is the proportion to which you have been equipped to speak words of comfort with wisdom.
I would also want to hear from those with whom my life is shared. One of the difficulties of coping with loss and grief is that it is with us wherever we are, including the workplace, the church, and other venues where our lives take place. Presumably there will be some in these venues who love me, and thus fall into the previous consideration. However, there are also the many acquaintances with whom we have frequent contact, but little by way of a loving relationship. It is important that these acquaintances don't draw themselves further away from those in grief, since the experience of grief is alienating in and of itself. To compound the alienation would only worsen the impact, I should think. However, I would think that an increased amount of intimacy should be avoided, on the basis of wisdom. Again, a bruised plant is vulnerable, and the acquaintance will be far less capable of providing wise care where those closer to those grieving can offer their love. I suppose I would want acquaintances to remain such, offering their condolences and continuing to treat me as an acquaintance without ignoring the fact that my life has been radically altered. I suppose I would also want a loved one to run "interference" for me by proactively seeking to communicate with acquaintances to let them know in what ways they can offer support and sympathy, so as to keep me from being inundated with attention, which can be wearying (at least for someone like me).
If you have other suggestions, or think mine could stand improvement, please leave a message. There is no better time to do so than there is right now.