I just did this for my wife. The iMac she got 10 years ago can't run modern games properly and isn't getting MacOS updates anymore. A Mac Mini with Thunderbolt video card + extra storage right now would be $1000-2000. I used http://pcpartpicker.com/ to help spec out an AMD Ryzen 5 build for $1000, that's very comparable with a topped-out Mac Mini 2018. Also stuck https://github.com/felixrieseberg/macintosh.js on there for nostalgia.
What parts did you choose? What display are you using?