Sorry for your loss when grieving and your in my thoughts for tough times.

I say, "I am so sorry. I will hold you close in my heart.? All you want to do is say something comforting - it doesn't have to be religious...and certainly not long.

Memories is all you have now cherish them and share them.

There are unlimited ways to comfort support and listen to the bereaved. ...religion is not one of them EVEN to a believer

My family has been dealing with quite a bit of grief, and one book that has been a brilliant approach to both dealing with grief, and to talking to someone who is grieving.
[amazon.com]

If I really care about the person I will say anything I think will make them feel better . Its a white lie big deal

yes i say sorry too

But if you are a nonbeliever, how you tell something like:" I hope find peace, tranquility and purpose." Great atheists don't believe in purpose. They are writers and atheists that don't believe something have a purpose and all ends with death. Do you believe that?

I say, sorry for your loss. You're in my thoughts.

I’m sorry for your loss.

I have found that an expression of compassion and empathy is useful. But in reality there is nothing I can say that will make it easier for the bereaved. However actions of kindliness are always helpful. Platitudes are about as useful as "tits on a boar hog".

If I was also close to deceased I try to find a small story to about the person. I also offer condolences and indicate that I am available to help should they need something.

My son asked me about this when someone he cared about lost someone close to them. It is tough because so many things that people say are unintentionally hurtful. I suggested he focus on his friend- “I’m sorry for your loss.” And “what can I do to help?” (Said genuinely with desire to provide support) and to be available to listen and empathize.

Your response sounds great

"I am so sorry for your loss!" What follows that I have to think depends on what thei bereaved has expressed. Some are focussed on their loneliness. Some on worry about how their loved one may have suffered, and others feeling some guilt feelings for some friction in the relationship that didn't get resolved or feeling they didn't do enough before the death. Just listening sympathetically can be the most important part.