Without knowing the OP's motherboard, it's hard to say exactly what his best move is.
Having said that, there are some things to consider:
1. Although an Intel build can provide a boost, that performance isn't always that high. Core i3 chips are fine budget builds, but don't always provide any benefit over the much, much older FX-83xx chips...& in fact, some games show them lagging slightly behind the FX chips. So an i3-based build might provide some 'future upgrade potential' by letting him move to an i5 or i7 build down the road, but isn't going to guarantee a noticeable improvement. And that leads to...
2. Price. Anyone who thinks the OP can simply replace his motherboard, CPU & RAM for the same price as any FX chip apparently hasn't been looking at pricing. The FX-8300 chip is running just over $100 USD right now (http://pcpartpicker.com/product/qmrcCJ/amd-cpu-fd8300wmhkbox). In contrast, a bare-bones Skylake i3 build with a microATX motherboard & the lowest-end DDR4 runs just over $200 USD, about twice as much (http://pcpartpicker.com/list/zb6zYr). But unless he has a case that only takes microATX motherboards, he's missing out on the expandability of a full ATX board...which will run him another $30USD (http://pcpartpicker.com/list/vtMf3C). And even assuming he keeps everything else (DVD drive, SSD/HDD, etc.), that also leaves him with...
3. Reinstalling Windows & every app. Changing out a CPU requires at most updating the CPU drivers for the system. Switching out motherboards, however, especially with switching from AMD to Intel (or vice versa), means you have to reinstall everything from scratch. If he's lucky, he has a retail version of Windows, so that he can reuse his old key; otherwise, he has to add another $100-120 USD on top of that price for a brand-new copy of Windows (since an OEM copy is tied to the original motherboard & is non-transferable).
The FX-8300 that the OP mentioned is not a "bad" chip; if it were, Tom's Hardware wouldn't be including it in their Top Picks (http://www.tomshardware.com/reviews/best-cpus,3986.html), & they even specifically mention how it's not only more affordable than a "modern Core i3", but has the added flexilibity of overclocking potential (assuming his motherboard can handle OCing). Plus, it's a much better CPU than his current one, being 3 tiers higher on the hierarchy (http://www.tomshardware.com/reviews/best-cpus,3986.html). With that CPU combined with the GTX 970, he's going to have more than enough computing power for gaming at 1080p resolutions at high quality (maybe even ultra, depending on the game).
Unless he's going to spend the time & money for a complete rebuild, an FX-8300 is a perfectly acceptable upgrade for him, especially if he's going to pair it with the older GTX 970.