I'm pretty happy with my 750 ti on an older 1440x900 monitor. Due to slightly substandard res it can do any game on medium. League maxed goes at 100+ fps, myself capping it at 120 fps that it goes in calm moments.
The 1030 is newer, tho slightly weaker option. Because new won't be as cheap. But at least you'd have warranty. It uses less power (750 ti is also very low power too tho) and therefor less heat. But watchout. Nvidia shadow released a stealth nerfed 1030 version with ddr4 memory and lower clocks (about 1000 MHz). Try to avoid buying that one. It's really crap. It'd be more proper to call it a 1010.
As for cheapo amd stuff options I'm less familar with. I sorta recommend one of those nvidia products tho. A radeon 240 would be much weaker than either of those I think, but it'd probably work just fine on league. AMD had really competitive cards back then. So although power hungry the higher end stuff is still effective. Hard to find a deal though. As for newer budget oriented amd stuff.. well seems to not exist I think due to crypto miners gobbling em up.
I feel like the 240 is both pretty old and low end tho. Even worse than a 1010. But idk maybe it's only 40$.
Are you building a new cheap rig or are you upgrading an older possibly prebuilt pc?