Definitely depends on the relationship first and foremost. I have learned the hard way that sometimes just listening for hours, being a shoulder to cry on or simply just sit with someone is all that is needed. I think at times that trying to be helpful can be misconstrued for a lack of compassion and when someone is drowning in grief they have little or no control over their emotions. Be there and do your best to be what they need, not what you think they need.
I can tell you what not to do/say... everything my husband is doing/saying during my grief. He continues to tell me that I am not being sensitive to his needs while I just lost my only sister with whom I was very close. He even left for two nights because he "needed to get away from me, didn't have a choice."
Do not be selfish, give the grieving person time and always be there for them. Even little texts throughout the days to let them know you're thinking of them. Sometimes the sadness will knock you down with incredible force, it must be nice to have someone to lean on during those times. I know that I wish I did. I can imagine that affectionate gestures would be helpful... I am considering finding those myself (not from my husband, of course.)