Boy, if there are two words I can't stand hearing together, it's "Stay strong." People say it all the time to cancer patients, but they say it to other people facing awful circumstances, too. It's almost always said with the very best of intentions, and when the speaker doesn't know what else to say in the face of the immense struggle the listener is facing.
Sometimes, what the speaker means is, "I wish you strength. I know you have to deal with something terrible, and I want to help you, but I have no idea how, because this is so huge. I'm hoping that my love, and the love of others, will help you tap some secret well of strength you didn't know you had, and you'll come through this awful time intact."
Sometimes, the speaker means, "Something huge and terrifying has happened to you, and it terrifies me, too. I don't know how I'll react if you fall apart. I'm afraid I'll fall apart as well. I need you to stay strong so I don't have to face that possibility."
Sometimes, people who say, "Stay strong," mean both of those things without realizing it.
Many hearers can take the phrase in the best possible light, and that's great, but I worry about the other people. What if they can't stay strong? What if they have used up all their strength just surviving to this moment and they desperately need to know it's safe to break down, that someone else will patiently wait and help them figure out how to put things back together again? What if you say, "Stay strong," and they hear, "Don't break down. Not here. I can't take it?"
Here is what I'd rather say: I'm here for you. I'll cry with you. Don't feel like you have to be strong for me, or for anyone. The only way out is through, and I know it will be hard. But I promise, no matter what, you'll never be alone.