From personal experience: wait a month or two, that's when the initial support dropped off and people expected me to be able to cope, yet this was when the shock had given way to the deep grip of grief - I had dealt with the immediate arrangements because I had to, after that was when I really needed help. Also you might want to ask first (dietary requirements /likes and dislikes).
And a couple of unsolicited tips -
Don't say "let me know if there is anything I can do" that puts the onus (and pressure) on your friend to ask {and it only took 2 "it's not convenient"s for me to quit asking anyone, too many people said it only because they thought they should}, Instead offer something more specific, if you're local you can say you're going grocery shopping on Tues, would they like to join /do they want anything picking up? You're in town for an appointment would they like to join you for coffee?
If you're further away, you could offer for them to stay at yours for a weekend, a spa day together, or a Haven caravan for a short break away from everything. Make the offer specific (the Xth, Yst or Znd of month) so they know it's a genuine offer.
Don't be upset/offended if they refuse though, it took me about 18 months to be willing to leave the house for anything other than what was immediately necessary. Keep asking, it's nice to know you're still welcome/wanted.
Don't ask "are you OK?" or "how are you feeling?" - I have a scar from actually biting my bottom lip to stop myself from screaming "HOW THE FUCK DO YOU THINK I'M FEELING?" how I was feeling was overwhelming, and while I knew the question was from genuine concern, it felt like they were stabbing me every time I was asked the question
Try something like "how's today going?"
One friend text me a photo of a dinosaur (in-joke) every single day: drawn on a post-it in pencil, in condensation on a mirror, in the frost on a car windscreen, a toy in the shop - it was a no-pressure way of letting me know she was thinking of me, and it made me smile, just a little, when not much else did. I had a photobook printed of those pictures and I treasure it more than I can really explain
If your friend does talk about their loved one, let them, even if it seems to be upsetting them, they may need to 'bleed off' the grief
Umm that's a bit of an essay, sorry