Lewis Howes explains why "let me know if you need anything" doesn't help grieving people. It puts the burden on them to ask for help. Instead, offer something specific like groceries, or just say "I'm so sorry for your loss."
Chapters
Best way to show for somebody who's grieving. Do not begin your sentence with let me know if. Let me know if there's anything I can do. It is disingenuous because what it does is put all the responsibility on you. Hey, I checked my box. I just said sorry. And I said, let me know if you need anything. I guess if they don't get back with me, they didn't need anything. And that's just not genuine. When somebody is grieving, the last thing they need is another burden put on them. Instead, you just do it. If you wanted to bring groceries, go bring groceries. If you want to make their life better, go make their life better. Otherwise, don't act like you do. It's much better to say, I'm so sorry for your loss.