You never really answered. Is her current FX 6100 blown too?
The difference between a FX 6100 to FX 6300 is small, sometimes virtually the same. Bulldozer to Vishera small architectural tweak. If her CPU is still intact it I would keep it and buy just a new board.
I wouldn't consider an i3 coming from an FX 6100. i3 may perform faster in most single threaded games but if you're getting 60FPS+ an extra few frames won't make any difference in the long run you might end up losing performance if her choice of games changes in the future.
The i5 seems like a good choice if the FX 6100 is blown, but also consider the FX-9370 8-core. It would give you better performance out the box. In games it performs anywhere from between the i5 2500k and i7 4770k depending on the game. On a dedicated chart its shows just 2-10 FPS behind the i7 4770k in gaming. In some non gaming tasks it can even compete with the i7 3960x which is a $1,000 Extreme Edition Chip.
Now the CPU itself is $220. So this would only work if you're willing to put in another $80 for the board, but you're getting i7 level performance on the cheap! - If this is an issue get the i5 or AMD FX 9370.
http://www.hardwarecanucks.com/forum/hardware-canucks-reviews/62166-amd-fx-9590-review-piledriver-5ghz.html