Zelda II was a good game, but it was frustrating. I feel like it wasn't the gameplay that made it frustrating, so it doesn't need an overhaul if it were remade. For example, the experience system was really neat and I wish it would be used in more Zelda games. The thing that made the game frustrating was that some aspects of it were unnecessary. So if I were remaking Zelda II, there would be some things I would add.

I would add Rupees and potions, and remove the lives system. A fatal flaw in Zelda II was the inability to recharge in places far from civilization. There was the Life spell, but if you ran out of magic, you were screwed. And the lives system was really out of place, and it was dumb that Link dolls were one-off items. I would replace all the lives with bags of lots of Rupees (and you'd get Rupees instead of an extra life at max level) In the towns you would be able to buy Potions, like Red Potions, Green Potions, Blue Potions, and Purple Potions. Purple Potions are in PH and ST and they act like fairies in other games. Of course, either Rupees would be a rare drop from enemies or potions would be a bit expensive. I always thought it was really weird to have towns without shops.

I'd also make the Second Quest a proper one.