I play Halo with my 5-year-old. I don't see anything wrong with that.

But I wouldn't let him play or watch something like Gears of War, it's too violent and there's horror elements. I remember being that age and watching black&white vampire movie and I was scared shitless. That stuff is ridiculously funny today's standards 😃

Splatoon is third person shooter but it's really great for kids also.

Well, I can tell you what my 10 year old was obsessed with - Roblox Nerf FPS - it's basically your run-of-the-mill online FPS, but since kids love Nerfs, it's...well, kid friendly, if shooting other kids with Nerfs can be considered friendly.

Also, Portal 1/2, though they are not really FPS, but he loves them.

Oddworld Strangers Wrath?!

Battlefield 1.

What kind of child are we talking about? A lot of times people think way too adult about games. While an adult might seem objectionable content the kid just doesn't give a shit because it doesn't get it. As long as it's SP most games should be fine.

Halo, Metroid Prime (it has fps elements deal with it).
As for TPS Splatoon and to some extent Ratchet and Clank and perhaps the first Uncharted. Team fortress also comes to mind and the old Jak and Daxter games.

I played Halo at a young age, the fact that you're fighting real aliens kinda helps (and it also helps you play as one in Halo 2, for empathy's sake).

Growing up I played Doom, Quake, Wolfenstein, Goldeneye, Counter-strike, Half-Life..you name it.

I'd say they're all kid friendly :P

Yoshi's Safari is the only one that comes to my mind right now.