" I am sorry for your loss". Specifically for your example.

They don't. They try & DO something!

Good question. I despise seeing that crap all over Facebook. I agree with you. Just something they say. How many of those folks actually stop what they're doing and pray?? Or, they use that little prayer hands emoticon. Yea that's thought provoking and profound.
Like that one church shooting last year. Immediately all over social media, people were sending their thoughts and prayers about the CHURCH shooting. Really?? Thoughts and prayers didn't exactly help there, did it??
Just empty words.......

People rarely offer to pray for me, maybe they think it's opening a door to (church lady voice) SATAN. When it happens, I acknowledge they're wishing me well...or maybe they're trying to justify their own delusions and validate their mythology by converting me.

I'll often tell people, I wish you the best or I hope you win the lottery.

I say. "What can I do to help you?" If local I offer to help with anything I can help with. A friend clear across country I ordered food from Amazon and a stuffed bear for them. Dont offer empty ideas. Do something.

I tell them that I am glad they knew love and hope they find peace.

We don't offer words, we offer action.

I offer condolences and then do what I can to change the situation

If they're going through or recovering from a medical procedure, "I hope [it] goes as quickly and as painlessly as possible." For deceased loved ones I offer my condolences.

Probably "Best Wishes"

What if you are thinking of them and don’t have a another word for prayers to show sincerity?

I say keeping you close in my thoughts and my heart. I like sending love and light, too.