Hey yawningmonster. I'm in West Auckland. Can I ask why you are hiring the car from Glenfield - it's a long way from the airport! How are you getting to Glenfield, is someone picking you up? If not then 0800 SHUTTLE is the big shuttle bus door-to-door company, you can prob just pick one up at airport, odn't really need to book. There are plenty of car rental places at the airport too.
I don't know the shops on the Shore very well but I do know there's a supermarket (New World I think) in Brown's Bay, which is just north of Murray's Bay. (I happened to be in Browns Bay a couple of weeks ago as they have an English food shop there - was picking up Jaffa Cakes!)
There is also a biggish Foodtown within the airport grounds so if someone is picking you up you could stop there.
From Glenfield I guess just head to the motorway, get on at Tristram Ave and it's only 2 junctions to Upper Harbour Highway, where you can come off for Murrays Bay. I know it looks huge and complicated on the map, but the traffic isn't that bad outside of rush hour and it's worse further down the motorway, towards the city - once you're as far north as Upper Harbour Highway the traffic has thinned out a lot
Coming back down the motorway, 2 junctions south to Tristram Ave for the hospital - I do know it is signposted on the motorway - at the right exit there is a big sign saying "North Shore Hospital - this exit" (Auckland traffic signposters get it right - shock!) It looks like you can avoid the motorway and get frim Murrays Bay to the hospital via East Coast Road but I don't know if it's any easier than motorway.
Honest, the traffic isn't that bad outside of rush hour. Or it doesn't seem so to me anyway. Hope your mum makes a speedy recovery.