I know people want to fix things, especially when it comes to grief. But you don't need to fix it. There is no fixing grief. The best thing you can do is just be present and listen.
Sometimes grievers need to repeat their story over and over. It helps them process their loss especially if it was a sudden or a traumatic death. Be patient. You may need a friend in the future to do the same for you. Be present. Showing up for them is very real for them. Suggest to go for a walk, a coffee, a chat on a bench at this time, at this place, make it a date and then all they need to do is accept or decline.
To say 'call me if you need anything' can sound quite empty, open-ended. Grieving people often have a hard time asking for help or making decisions, so it's up to you to reach out. Or not. It depends on your relationship with them. But if you have thought 'maybe I should give ... a call', then do it. You won't regret it. It may not be an easy conversation, but when in life were things meant to be easy.
Long after the funeral grieving people will still need you. Check in with them. People always make the mistake of assuming how the griever may feel. Unless they have told you to give them space they probably would really appreciate a call.
Sometimes quite a bit of time has passed since you connected them and you may feel that you've blown it. What is the worst that could happen if you did contact them? They'd tell you off? Well at least then you'd know you blew it. You could start by saying 'I've been a terrible friend, but I didn't know what to say'. Honesty always wins.
The short of it is. Unless you are an incredibly rare human, we will all encounter grief. And we will not know how it will feel until it happens.
So until then go gently and kindly, with love and patience. And reach out!