That's actually an interesting question with a subject that is really under talk!
I'm also really careful with copyright and intellectual properties. But first one of the rules is that as long as you don't make money it's safe to copy (kindda sad but true). Nintendo won't sue you for making a remake of Mario if it's for yourself or just for some small range diffusion. It will cost them more to start a trial against you than what you will be fined if so. Same thing goes for Fan art.
Also, Space Invaders is such a cultural symbol that somehow don't really belongs fully to it's creators (Atari?). Adding the fact that it was published so long ago that the patterns and copyright must be royalty free by now.
How many copies of the arcade classic do you see everywhere? Candy Crush is a copy of Columns, and Angry Bird is just a more refined version of a old game called Tanks, that was maybe a copy of something older.
If you do a Space Invaders just name it Alien Invasion or something and change the shape of the main enemy by two pixels and that will be ok.
The biggest risk will be that Atari, or whoever has the remaining rights to the original Space Invaders game, ask you to take your game down from the gaming platform it's on. But then they will need to take a whole lot of game too
The notion of copy or in this case intellectual property is a really complicated one. Was Crash Bandicoot Racing a copy of Mario Kart or just a rip off? Technically Nintendo introduced the Go-Kart Game, but is completely unable to copyright it. So you should not really worry about paying royalties