Really if $50 is your limit a 740 is the best bet. Contrary to what people on the forum here say it will run modern games ok including the lego ones - people on this forum are obsessed with super high res and ultra settings. I know someone who has one and he doesn't seem to have any problem playing games on it.
As a step up from that a 750Ti is also a good card - plenty performance and ticks all the other boxes (small, low power, modern architecture, good drivers).
The radeons are much bigger, hotter, much higher power and will need more maintenance. Second hand is asking for trouble.
If I were buying something like that for my son's pc I'd just get the nvidia card, turn on geforce experience so it sets all the game settings up appropriately and know it will just work without me having to fiddle with it every other week.
As a step up from that a 750Ti is also a good card - plenty performance and ticks all the other boxes (small, low power, modern architecture, good drivers).
The radeons are much bigger, hotter, much higher power and will need more maintenance. Second hand is asking for trouble.
If I were buying something like that for my son's pc I'd just get the nvidia card, turn on geforce experience so it sets all the game settings up appropriately and know it will just work without me having to fiddle with it every other week.