@Drizmod 1060 6gb is better for older titles, rx580 8gb is better for newer titles, on average, they're neck and neck
the benefit of going rx580 is that freesync monitors are cheap and plentiful, whereas g-sync monitors still cost an arm and leg to buy in comparison.
so imo, going mid-range cards, 580 8gb is a no brainer really.
however, mid-high/high-end cards, there's only nvidia at the moment, as amd's offerings are currently lol-tastic (but your budget doesn't allow for that without compromising on other parts)
ps: is the monitor and peripherals in a separate budget??
i just re-read your opening post and it doesn't look like your friend has a monitor/kb/mouse either, and none of the builds here have specced in those
if he was using a 144hz screen at 1080p then gtx 1070 would be a better choice, but greater costs .
nice !