Have played Tomb Raider for PSX? It work very well, but costs a item to save. This shows how to do it right.
Are you sure? This kind of a system would be far too arbitrary to work well with a game like LoD.
It actually ties in with your next concern:
The problem is that even right it can backfire on the player, since you can save with low health and no items, meaning that sometimes you will need to restart the game from 0 (its rare, but happened at least 1 or 2 times with us here).
Exactly. And it'll probably happen more often than you might think. If you're low on health, or find yourself in a difficult situation, it might be a critical mistake to save your game -- you'll often be forced to start over entirely. And yet, when we're given that option, that's precisely what many of us do; it's compulsive, a survival reflex. So implementing this "free" save system isn't the best of ideas.
Now, of course, there are ways to circumvent that; think of what HoD did: you could make a "quick save" in any room, but only save rooms allowed you to regain health and magic. And when returning from the title screen, you could choose to return either to your quick-save point or to the last save room you'd encountered -- so that you won't be forced to bend to that hasty bit of checkpointing after all...
But this, of course, could really come to the detriment of a decent level of difficulty. For LoD in particular, I don't think it would be a good solution.
So here's a last possibility: along with the traditional save feature, a "sleep" mode that will essentially save your game at any point during gameplay, but that will forget your progress until you choose to put the game to sleep again. This wouldn't allow cheating, since you can't quick-save, continue your game, then revert back to the save if you aren't satisfied -- the save would simply no longer exist. I think AoS had something similar; it's simply a way to avoid losing progress if you have to leave without having yet reached a save point -- essentially a mechanism to facilitate coming back, and nothing more. And I think an auto-save feature would work well with this too.