When my mum died, one of the most helpful things a dear friend did for me was a big shop for food and drink I could put out with people coming to the house. Left on the doorstop was wine, soda water/soft drinks, cheeses, chips, dips, biscuits, tea, etc. Total surprise and took the pressure off.
Just on providing cooked food, lovely idea but we ended up having to throw most of the food away.
The other big help was the day of the funeral, 3 very close friends of mine offered to organise the wake. They left the funeral and went straight to my house to let in the caterers and get everything set up. World of difference not having to think about logistics on the day.
About offering support, this is a hard on as everyone grieves differently. My only advice is try not to put the work on your friend; instead of let me know if you want to catch up, try I'm going to x cafe on x date at x time, I would love to shout you a coffee if you feel up for it.
And finally, try not to be awkward when your friend mentions their name, and make sure you feel ok in saying their name. It sucks when people get weird.