I'm not able to relate the grammatical structure behind the following to any of what I’ve learned:
etwas + [adjective]-es
What is it called? Is it a special structure exclusive to etwas?
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It has to be etwas Besseres, as the adjective is made into a substantive here. This also works without etwas, so there is no special grammar structure behind this.
Ich habe Besseres zu tun.
I have better things to do with my time. I have something better to do with my time.
English needs the thing as far as I'm aware of. German leaves thing out. Though you can write, gramatically correct,
Ich habe bessere Dinge zu tun.
that one sounds stilted. Something literally translates into etwas, but German doesn't need the etwas here. Maybe that's the source of your confusion. German speakers will always assume etwas to be a small amount. So the quality changes when etwas is used:
Ich habe etwas Besseres zu tun.
I have some very tiny, yet better thing to do with my time.