Well this is how I look at things, when budget is of concern (especially on the low end):
Intel options:
* i3 is pretty good for 95% of duties. Might have some issues if you have multiple things going on during gaming, but generally is pretty good.
* Non-OC'ing Mobo's are quite cheap, if you don't need too many USB ports and/or SATA connections (there are exceptions, but generally the rule).
* Does consume less power, but usually isn't too much of a concern within a desktop environment.
AMD options:
* CPU is very well rounded for pretty much any task you throw at it, especially if you haven multiple things going on (i.e. gaming while listening to music, while having multiple windows open, etc.).
* Mobo's are usually a bit more $ than the cheaper Non-OC'ing Intel Mobo's, but they generally have a decent amount of SATA ports (usually 4 to 6 or more) and usually more features for comparatively priced Intel Mobo's (of coarse there are exceptions).
* CPU is also OC'able, but will need optional CPU HSF to push the limits of the CPU OC, which means about $20-30 more (which means your CPU price difference is about equal in initial costs).
The other thing to consider is that most people will probably not upgrade the CPU down the road, because there is usually some other compelling options that they might take advantage of, like SSD's or something like that. Obviously if gaming is a primary option, they might need to upgrade to a 4 core CPU down the road to meet a "New" game requirement. You get the idea here.
Personally I think the FX 6300 is a good all around CPU to have and might just be a slightly better option over the i3, but that is just my opinion. I think if you go with the i3 option, the owner might want to upgrade to an SSD down the road, instead of a CPU upgrade, but that will depend on what the owner wants to do to improve their system (the same could be said about the AMD option too).
Examples of possible builds with either CPU/Mobo needed:
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel Core i3-4160 3.6GHz Dual-Core Processor ($119.99 @ Newegg)
Motherboard: ASRock H97 PERFORMANCE ATX LGA1150 Motherboard ($74.99 @ Newegg)
Memory: G.Skill Ares Series 8GB (2 x 4GB) DDR3-1600 Memory ($72.00 @ Newegg)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($59.99 @ Newegg)
Video Card: XFX Radeon R9 270X 2GB Double Dissipation Video Card ($159.99 @ Newegg)
Case: Corsair 200R ATX Mid Tower Case ($59.99 @ Newegg)
Power Supply: Corsair Builder 600W 80+ Bronze Certified ATX Power Supply ($29.99 @ Newegg)
Total: $576.94
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2014-10-12 23:50 EDT-0400
PCPartPicker part list /
Price breakdown by merchant
CPU: AMD FX-6300 3.5GHz 6-Core Processor ($109.99 @ Newegg)
Motherboard: Gigabyte GA-970A-UD3P ATX AM3+ Motherboard ($74.99 @ Newegg)
Memory: G.Skill Ares Series 8GB (2 x 4GB) DDR3-1600 Memory ($72.00 @ Newegg)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($59.99 @ Newegg)
Video Card: XFX Radeon R9 270X 2GB Double Dissipation Video Card ($159.99 @ Newegg)
Case: Corsair 200R ATX Mid Tower Case ($59.99 @ Newegg)
Power Supply: Corsair Builder 600W 80+ Bronze Certified ATX Power Supply ($29.99 @ Newegg)
Total: $546.94
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2014-10-12 23:52 EDT-0400
Tried to get similar options on the Mobo's, so you could see what kind of pricing you can see. Obviously there are several MIR options, so you can factor those in/out if you like. Both Mobo's have a Front USB 3.0 header on them, so you can take advantage of that on the Corsair 200R. Both Mobo's are ATX and come with 6 x SATA 3 ports.