Going back to my hippy days of the 60s - Peace and Love!
When writing I oftern say - Wishing you all good things.
On Facebook - Hugs xx

I'll keep my fingers crossed

Peace be with you

"You'll be in my thoughts" is my go to, along with, "do you have my number/messenger?"

Vibes.

My condolences

{Name} , we are grieving with you.
More:
Our hearts go out to those affected by ___.

When it comes right down to it, an equivalent expression presupposes not simply a higher power but a known and accessible one, which means you could not be an atheist or agnostic and have an equivalent expression. Even for a non-theist, because the concept of praying for an individual contains within it a request, the closest you could come to that expression would be that you would chant an incantation or utter a spell, which I imagine would be similarly off-putting to you and the individual you're interacting with.
I think your approach was likely the best. You could include a statement like, "Let me know if there is anything I can do to help," which you could even preface with an admission that you're not religious if you feel so inclined.

"Let me know if there is anything that you need, and I will try to do it." or anything along these lines. Actions are much louder than words.

Sending thoughts your way.

I believe that is "good luck"

What I've learned is that we're all human beings with hurts and pains and in need of kind words from others. I told a very good friend that i see one to two times a year, God bless you. Why? I know his conviction to his God. Whats a little white lie to a friend that wants to know im doing ok in his world of loving and helping others. I have nothing to prove to others. There are consequences to being over zealous and boasting about who and what i am to persons who wouldn't understand. My idea of helping others is to not control them with my beliefs. When people ask me why I'm nice and or caring to others. I say love others. Its the right thing to do.

Ask them if they need help financially. Offer any assistance that might help take some of the burden away so they can focus more on their loved one's care.

It is an empty statement to say "I'll pray for you" but at the same time it is reassuring to those that want to hear it. It irritates me to get "prayers" because it feels like a cop out so they don't actually have to do anything or have any real advice or lend an ear.

Wouldn't it be better to offer help, maybe express your concerns for their issues, let them know how sorry you are and just listen? To me saying that you'll be thinking of them is equivalent to offering prayers but that might be what they want to hear. Telling them that is sad, expressing true regret about the situation and listening might just be all that can be done.

I always say I wish you the best or my thoughts are with you.

Man or Woman up! Seriously the invocations of prayer can be verified in the quantum world where as I understand the various sub-atomic particles can be seen to re-act whilst a bunch of Scientists concentrate on the particles willing them to move.They do move and this phenomenon is yet to be explained by Newtonian and Einsteinian Physics! Therefore my four word admonition at the start of this post holds firm at this stage until we have a satisfactory explanation of the concerned particles specific actions!

Good question..

I'll send you good vibes.

I,m thinking of you or you'll be in my thoughts seems to work

"Let me know if there's any way I can help you and your family" is my usual response at a time like this, especially if it's someone close enough to me that we're having this conversation in the first place. It's something more concrete than "thinking of you" and definitely more productive than prayer.

What i say in those kind of situation is, ''j'ai une pensée pour toi'' , a traduction would be , ''i will have a thought for you'', meaning i will take some time to think about you in the next days... I don't know if that's something you guys say in english

To me the answer is simple and straight forward. I just tell people that I will be thinking about them as they face whatever the issues might be. Sometimes I say," You will be in my heart and in thoughts." I have never had anyone ask me if I would also pray for them, and if they did I would say to the person, "I am sure that you have others that are taking care of the, but I am giving you the best I can straight from my heart and thoughts."

That is a hard one - there is really no replacement you can offer as atheist for getting a little magical help from heaven.

Like DeeTee, I will say "You (or they) will be in my thoughts," or "I have been thinking of you." But if you can, try to think of some concrete way to be of assistance. It would have been difficult to come up with anything in the particular scenario you mentioned, but sometimes in other situations bringing by a casserole at a time of grief, or offering to look after kids while someone spends more time with a sick relative, anything like that - can get us around the "I'll pray for you" thing and show what active care is about. At work you might be able to take on a bit more of a common project so that someone who is suffering personally can deal with whatever issue at home.

Best one I've seen is: "I'll think for you". A little snarky, yes, but it does get the point across.