Sorry bit of a long one
hey I'm currently running a homebrew game and I have 1 more player joining in for the next session.
I'm looking for some input for this characters backstory.

The 'Part' of the backstory that I have full control over:

The first memory you have of the island you are on, is you were just walking around not really aware of what you were doing. until you snapped out of your day dream, you have no idea of how you got there. You started to explore the island and eventually found a small spring (the only water source on the island) you spent some time on this island, but your not quite sure how long, it feels like you've only been there less then a week. But deep down you know you have been there a lot longer than that. your memory of the island all together is very hazy. at some point while walking around the island you start to hear a voice inside your head. At first it is very quiet and hard to understand, but you manage to make out "Ren... Ren, is that you?" replying with "yes.. who is this?" Immediately after a portal opens up in front of you and a someone walks out towards you.

The spring that that pc finds grants immortality, in the sense that when the pc dies they will be brought back to life, no invulnerability.
Question - in the instance the pc dies, what's the best way to bring them back alive without it being obvious. As the player doesn't know about this

I have talked with all of my players about throwing something overtop of their race and class giving them special powers and so on and they welcomed it.
And I'm not cruel, I'm not going to force this player to play this character forever, if they do want to kill it off I will wrap the story up for them.

I will tell them straight what is happening with their character at some point, but I will try to keep the cloud of mystery as long as i can.
bbeg drains the full life force from something and then tries it on the player and doesn't work.
meets a god and they start questioning what they are as they are not natural.
things like that. If you have any cool ideas to help feed that cloud, that would also be awesome.

Any input is greatly appreciated.
Thank you

The most common way a character dies in 5e is reaching zero hit points and failing 3 death saves. The player is completely aware of that entire process. They know the character died, the other characters also likely know they died, the character themself probably thinks they died. So when you say "No you didn't die", everyone knows something is going on. All of your examples were for cases where the DM could intervene or change the rules for the one character - life draining attacks/god interactions.

However, if the character learns they can't die, the odds are good that they will start taking risks, doing ridiculous things since even though the chance of success is small, the cost of failure is negated because they can't die. There are a number of science fiction and fantasy novels about someone who discovers that they can't die.

Anyway, there are a couple of options - they could be immortal in the sense that they will not age, will not die of diseases or other afflictions, will remain healthy ... so they could regenerate major injuries over time with significant downtime and recovery (In this case, perhaps if they die, their heart starts beating again after a few hours and wounds start repairing themselves over a few days to weeks - their body doesn't start to decay as if under the effects of a gentle repose spell). Alternatively, for an instant recovery, they could have a limited number of lives and the character doesn't know how many. Some power granted them more than one but they never know which will be their last so dying will never be a risk taken lightly.

So with reincarnation you can pretty much do anything - change race, gender, age etc - maybe it’s a curse, maybe it’s a spell.

if the character dies have your player roll a new one - they don’t have to know it’s the same being.

maybe the maguffin creates false memories to protect its discovery. 

there’s a character in DC called resurrection man, every time they die they are reborn again with a new power.

Sounds like you need a "Magical Feather" (re: Dumbo).

Basically, have an item, maybe an heirloom that grows warm when the player fails their 3rd saving throws. The item is just a placebo, but nonetheless, it coincides with a Spare the Dying type effect that "stabilizes" them. (Avoid doing Instant Death damage).

When you're ready to reveal the truth, have the character lose the item and when they die and still recover, they'll realize "it was within them the whole time".

As for the "air of mystery", maybe have each death extend a narrative. Like, the first death, they find themselves in a void for just a moment. The second death, the void last a little longer and they start to explore. The third death, they wander far enough to find an ornate gate. On subsequent deaths, the door begins to open just a little and voices can be heard on the other side.

Basically, each death gets them a little closer to the "Other Side", whatever that means.

I'm actually dealing with a very similar situation right now. My player is aware of the ability that he has, but he hasn't asking risky because that not what the character would do and also out of respect for me. When he does find out make sure to set those boundaries with him right away that even if he is immortal, he doesn't know the nature or reason why so its still very dangerous to risk dying. 

Fortunately for us, there are several classes that deal with life after and outside of death currently. Zealot Barbarian, Phantom Rouge, Shadow Sorcerer, Undying, Undead and Celestial Warlocks, Grave Clerics (Kinda) Probably others too!

What I have done is give my character a single, automatic stabilization when he falls unconscious in combat. This happens ONCE per life of his. While he is resistant to death, he is not immune from it. Also, when he does die and come back to life, it is after several in game weeks and they steal the body of another individual in the world, transforming it into their form and have to deal with having another person that is always with them. The player loses a large chunk of their memories every time it happens, remembering only their name and the strongest of goals/emotions (He is a cleric so he doesn't level down or anything). This memory can only be recovered by narrative actions and not spells, save probably a wish spell because its wish. The implication of this is the guilt of having to deal with robbing another person of their life and also as of yet, never being able to know when they have gotten all of their memories back.