Like the title says, I have a player who wants to remove his own body parts to replace with that of fallen foes to gain "abilities". 

A few points:
1) Campaign Setting - Icewind Dale Rime of the Frost Maiden
2) PC's current level - 6
3) Max Campaign Level - likely 13
4) Both I and players are open to the idea. 

I'm ok with the consept. It also fits with the theme I've been cultivating of "the slow decent into madness".

A little, background the player is a human Reghed Tribesman Rogue who thinks that, for example : "if an orc can see in magical darkness, its because he has magical eyeballs and wants to remove one and  replace his with it."

Any ideas on what could be a fun and balanced system to implement this?

My first thought is a -1 to constitution for each "augment" he does? maybe to represent the trauma his body is undergoing... However, I'm not sure a good way they could attach one and get the benefits. They do have access to a 2nd lvl Cleric of Shar in the party. 
They are headed to the Dark Dutchess soon. Maybe among the hoard they find a Tomb on Flesh Golemns or somesuch that they can perform as a ritual?

Ultimately, it seems like it could be a fun and memorable experience to explore, and I'm just trying to get some ideas that can make it just that for everyone at the table.

Actually, a magic item is probably the way to go. But you'd probably have to homebrew something or just use a Manual of Flesh Golems.

I’d use re-skinned magic items. In the example, goggles of night would fit. The big advantage they’d have over traditional items is you can’t lose them. But in practice, I find characters rarely lose items anyway, so realistically it’s not much difference. 
If you want to out a throttle on him doing it, I’d say have it take an attunement slot. 

I would require an attunement slot for the first item.  Not for future ones.

If he did 3 or more exchanges, I would change his type to monstrosity.

Also, -1 penalty to his choice of Constitution or Charisma would be a risk every time he did a replacement. Give him a Con save with an increasing DC, starting at DC 14, +2 every already existing transplant.

Make less simple to attach the limbs, make a smallish side quest that involves the player possibly going to find a way to attach them without harming himself in the process, or even take a long winded detour to some strange and slightly crazy wizards tower to be able to attach them. Another way would be to use some sort of cursed or otherwise strange spell that will give him effects later on, make it so the limbs will turn a strange gray shade or begin to have his veins turn darker. Then maybe take that and give him a penalty of being not only a monstrosity (Mog_Dracov) but let him be horrifying to see, disadvantage on all charisma and cannot fit into certain armor. But of course depending on what he wants to add in terms of bonuses you could also grant racial traits or other things such as darkvision. Hope some of these ideas help. 

Coincidentally, Pathfinder 2e just came out, like literally yesterday, with "Howl of the Wild" which is sort of its "feral" book.  There's a whole system of grafting bestial body parts to take place of an adventurers born parts (like insectile compound eyes for superior vision, actual claws, etc.).  I know there's PF2 > 5e conversion systems out there, but don't know any enough to recommend it, but if you really wanted to do this, you might want to check it out.  One thing I did like is they were treated in many ways like magic items in that particularly powerful grafts did take the equivalent of an atunement slot.  Think the PDF would set you back $20.

EDIT: oh yeah, Fizban's has an examples of dragonflesh grafters, sort of experiments performed by the cult of the Dragon, grafting dragon parts onto humanoids.  Might want to look there before jumping over to what a different system does.  The examples are monsters, but might give you an idea on how to scale the techniques for a PC, and maybe some consequences.

"I wanted infravision like the elves... But 'tis more than just taking their eyes..." ~ Xzar, Baldur's Gate.

I'd implement a system where he can attempt it but he has to roll a nat 20 or a 100 in order to do so. Most of the time it shouldn't work but there might be the rare occasion that it will. So keep on trying! 

I'd also go with it uses attunement slots. Maybe give him a total up to his constitution modifier if you want to be generous. He's basically a Doctor Frankenstein version of an Artificer at that point. 

Using Reddick as an inspiration point, i could see a human getting a magic surgical enucleation to get eyes replaced with those from nocturnal creature and get darkvision. One of the downside could be disadvantage to Charisma check during social interactions with said creature's kind for example. Or progressively develop a Sunlight Sensitivity trait. If the character is frequently making attack rolls., i'd consider a variant Minor Sunlight Sensitivity that only avvec Wisdom check.

Sunlight Sensitivity. While in sunlight, you have disadvantage on attack rolls, as well as on Wisdom (Perception) checks that rely on sight.

I think this is brilliant and the con aspect is one poster noted doing it as a negative to the save. Could you imagine Rastlin swapping out his legs for that or so he was stronger and faster changing out his own eyes swapping out an arm. This could be so cool.

Seems awesome, but I think they also gain the creatures weaknesses

I'd treat this as another form of magic item, requiring attunement. You could even consider making them tiered, like the "horde agic items" from Fizban's, to reflect new changes as the extend of the modifications increases. I would also start with the artificer's infusions for inspiration, since many of them can be re-skinned as grafts and whatnot.

Quote from Riderstrong >>

Like the title says, I have a player who wants to remove his own body parts to replace with that of fallen foes to gain "abilities". 

<snip>

I'm ok with the consept. It also fits with the theme I've been cultivating of "the slow decent into madness".

 
Hah, in my book that's more of an out-of-control train car plummeting towards the depths of madness at breakneck speed.

you would need to

1. remove the body parts you want to replace
2. place the replacement where you want it
3. cast a healing spell on the wound with the replacement attached

you said slow descent into madness? I got you fam.

1. every long rest, very obviously roll 1d12 (the scariest die) and make note of the result. DO NOT tell them what its for
2. if the roll is under 6, their mind gets assaulted with memories of creatures they have parts of while they sleep (you said orc eyes? they see the image of everyone the orc had ever killed)
3. they wake up with 1 level of exhaustion