Update:

So I called the Hyundai dealership first and gave them the VIN and all and they said that there was no engine recall for our car. I also looked it up myself and it didn't show a recall for it.

Today I called my mechanic that I've been using since I was 15 who knows what he's doing. I told him what was going on and he said that he'd bet it's a timing chain issue. That could mean that the timing chain is just badly stretched, a timing chain guide is broken (they're plastic), or a timing chain tensioner isn't working properly. I used to have an '05 Mustang GT 4.6L that had a similar rattling noise and come to find out, one of the timing chain guides (also plastic eyeroll) had broken into pieces and the timing chain was rattling up against the timing cover.

So when I got home from work I tore into the car a little bit.

I've got the valve cover off and the car is up on ramps. Tomorrow I'll work on getting all the accessories off of the timing cover and then maybe get the timing cover off also. We'll see. I'll update when I make some more progress.
I did notice when I took out the spark plugs, which I just replaced a couple months ago, looks pretty rough and burnt sort of. I'm not sure what the deal is there. Maybe it's related?

 

Update:

So tonight I got back to work on the car and I was planning on getting somewhat close to taking the timing cover off. I discovered that it's pretty much impossible or close to it to take off the timing cover with the engine still in the car. There isn't any space whatsoever to get a tool on the bolts on the timing cover and much less a pulley-puller tool to take off the harmonic balancer. I would totally pull the motor. In fact I'd enjoy that a lot, but I don't have a cherry picker to do that with. Even if I did have a cherry picker, my driveway is gravel, so that wouldn't work.

I guess I'm gonna take it to my mechanic and see what he can find out for me. Some good did come from me wrenching on the car. I found that the valve cover gasket needs replacing anyways. It's leaking oil out of the back driver side corner of the valve cover. Also, the serpentine belt on the car is very dry-rotted and cracked. So I'll be doing some needed repairs on it anyway.

I'll post again once the issue has been resolved with some answers.

 

Hi! So did you resolve it?