Best MBO for Gigabyte GTX 1050 Ti Windforce OC 4GB GDDR5 and AMD Ryzen 5 1600?

Ryzen 5 1600. Sweet spot for price/performance. Includes a beefy stock CPU cooler in the box (the 1600X doesn't).

You need to get new RAM as only DDR4 sticks are supported by the Ryzen (and even current-gen Intel). Sell your DDR3 to offset the costs for the new parts. Ryzen likes high-speed RAMs, so, you want to look at some 3200MHz 2x8GB kits.

As to the motherboard, depends on your needs, your case size, and your budget. A B350-chipset mobo will suffice for most general needs (including overclocking, whether you intend to or not).

To give you an idea, here's a suggested parts list for the CPU-MB-RAM (I don't know if you have other components as well, like GPU, HDD, SSD, PSU, monitor, case, etc., so, they were not included):

PCPartPicker part list / Price breakdown by merchant

CPU: AMD - Ryzen 5 1600 3.2GHz 6-Core Processor ($195.89 @ OutletPC)
Motherboard: MSI - B350 TOMAHAWK ATX AM4 Motherboard ($70.98 @ Newegg)
Memory: Corsair - Vengeance LPX 16GB (2 x 8GB) DDR4-3200 Memory ($158.99 @ Amazon)
Total: $425.86
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-10-03 13:23 EDT-0400

For the PSU, your rig with a GTX 1050 Ti can be run on a ~350W PSU; with a GTX 1060, a ~450W PSU. But good-quality and great price/performance PSUs that are available are usually in the range of 450W-550W.

Getting a 450W PSU will have more than enough juice to power your rig, whichever of the two GPUs you pick (or even a GTX 1060-6GB). A 550W PSU will give you ample headroom should you choose to upgrade to more powerful components in the future (e.g., GTX 1070 and above), without having to change such PSU. It's up to you which to go for, depending on your budget at hand:

Recommended good-quality 450W PSU: Corsair CX450M (2015 edition)
Corsair - CXM (2015) 450W 80+ Bronze Certified Semi-Modular ATX Power Supply ($26.99 @ Newegg)
Note: That's ~$27 after $20 rebate at Newegg

Recommended good-quality 520W PSU: Seasonic M12II-520 EVO
SeaSonic - 520W 80+ Bronze Certified Fully-Modular ATX Power Supply ($52.59 @ SuperBiiz)

Recommended great-quality 550W PSU: EVGA SuperNova G2 550
EVGA - SuperNOVA G2 550W 80+ Gold Certified Fully-Modular ATX Power Supply ($88.83 @ Amazon)

For the GPU, both the GTX 1050 Ti and the GTX 1060-3GB are meant for mid-level 1080p-resolution gaming. The GTX 1050 Ti, despite having 4GB of VRAM, will still perform less, i.e., lower fps (frames-per-second) compared to the GTX 1060-3GB, due to the lower number of CUDA cores (only 768 cores for the GTX 1050 Ti versus 1,152 cores for the GTX 1060-3GB), not to mention, lower clock speeds, memory bus/bandwidth, and power.
[video="https://www.youtube.com/watch?v=6-zXJJz05Co"][/video]
Although such 3GB-VRAM GTX 1060 GPU will produce much much higher fps than the 4GB-VRAM GTX 1050 Ti, especially on 1920 x 1080 resolution, the lower VRAM might be a limiting factor on selected games that uses a lot of it when graphics settings (eye-candy) are enabled, especially on higher resolutions. The 1060-3GB may still produce much higher fps, but, the game might not run smooth if more than 3GB is needed (in such high graphics settings).

This is why most opt to get the GTX 1060-6GB version instead of the 3GB, provided, the price between the two GPUs is close. Note that the 6GB version does have twice the VRAM of the 3GB, but, it also has the same clock speeds, memory bus/bandwidth, and slightly more CUDA cores (1,280 cores). With its higher VRAM, it is very ideal for high-end/top-of-the-line 1080p-resolution gaming with higher graphics settings enabled. However, the GTX 1060-6GB won't produce as much as a wider gap in fps versus the GTX1060-3GB (compared to the GTX 1050 Ti).
[video="https://www.youtube.com/watch?v=yleFZMFgabk"][/video]

Choice of the GPU will depend on the games you play, at what resolution, at what graphics settings, the fps acceptable to you, and your budget.

In short:
GTX 1050 Ti --> $150-$170 budget. Medium settings (in most games) at 1080p to achieve high fps.
GTX 1060-3GB --> $220-$240 budget. Will produce significantly higher fps than the GTX 1050 Ti.
GTX 1060-6GB --> $270-$320 budget. Slightly higher fps than GTX 1060-3GB at much higher graphic settings.