The Bad News
As others have pointed out, Invisible Stalker is a monster and the stat block it uses is just not viable for a PC. You could always start tweaking the different parts of it to try to get something balanced but chances are it would never really pan out and your player would still be far too strong, or so saddled with drawbacks that he can't have any fun. So the idea of playing an Invisible Stalker is probably just not going to work...
The Good News
Unless your player is willing to work with you to come up with an in game reason for why they are invisible. If the PC was a standard race and the invisibility was a side effect on top of that, you don't have to worry about balancing anything other than the strengths of invisibility. And since we are firmly in homebrew territory anyway you can play around with how the invisibility works and what drawbacks you want to give it to make it more balanced.
The Invisible Man
As a really rough idea, let's look at the classic Invisible Man. Wrapped in bandages from head to foot, wears a long trench coat, has a hat and shades on even indoors. Why does the Invisible Man look this way? Because his invisibility doesn't affect his gear. Unlike the normal invisibility spell which will also hide items you wear and carry, whatever gives the Invisible Man his power is only skin-deep.
When the Invisible Man wants to sneak past a guard, he has to lose all of his clothing and sneak around naked to do so. If he gets found out while sneaking in this way he is going to have to deal with that situation with no armor, no weapons, and barely any dignity. Being magically invisible during a fight is a pretty big advantage. Being the Invisible Man in a fight means you are running around naked swinging a sword (that everyone can still see floating in mid air).
What made the Invisible Man the way he is today? Maybe it was an ancient curse picked up early in his adventuring career. Maybe it was an alchemical experiment gone terribly wrong. Maybe there is an actual Invisible Stalker lurking somewhere in his family tree. The reason why is something you should work with together with the player or even let them come up with on his own. In my experience, creating a backstory for a cheesy character like this is half the fun of playing them. Plus you can get a free plot hook out this, if the PC is trying to undo whatever made him invisible to begin with.
The Unreliably Opaque Man
A slightly different take on the same idea above, but with the invisibility not being a constant effect on the PC. Instead let the invisibility be something mostly out of the player's control. Figure out some trigger that you think would be fun and then have the player roll every time the trigger happens. If they succeed (or fail, depending on how you determine the roll) then they turn invisible.
As the PC levels up, maybe let them start to control the invisibility and attempt to turn it on even without the trigger. This represents them coming into their power, with the end goal of being able to fully control their visibility by level twenty. Of course, most games will never go long enough for PCs to hit epic, but even if you wrap up before then you can give your player a little epilogue note saying that the PC reached his goal eventually.
Have Fun With It
Letting your player just walk around invisible all of the time, with all of the bonuses that that implies mechanically is clearly not feasible. But if you are both willing to work together there are options for a PC that can less visible than most without breaking your whole game. Make sure the other players are okay with it as well, since you don't want to exclude other people just because of one goofy idea, but if everyone is onboard I would at least give it a try. Remember, D&D is all about having fun and telling stories together. The rules give you a box to do that in, but sometimes you need to stretch that box out to fit your idea.