Well the cheapest BYO I could come up with, just over $600 (~$670 w/ rebates)
PCPartPicker part list:
https://pcpartpicker.com/list/qGmLpG
Price breakdown by merchant: https://pcpartpicker.com/list/qGmLpG/by_merchant/
CPU: Intel - Core i3-8100 3.6GHz Quad-Core Processor ($109.99 @ Newegg)
Motherboard: MSI - B360M PRO-VDH Micro ATX LGA1151 Motherboard ($67.99 @ Amazon)
Memory: G.Skill - NT Series 8GB (2 x 4GB) DDR4-2400 Memory ($70.98 @ Newegg)
Storage: Seagate - Pipeline HD 1TB 3.5" 5900RPM Internal Hard Drive ($39.99 @ Newegg Marketplace)
Video Card: MSI - GeForce GTX 1050 Ti 4GB Video Card ($164.89 @ OutletPC)
Case: Thermaltake - Versa H17 MicroATX Mini Tower Case ($24.99 @ Newegg)
Power Supply: Corsair - CX (2017) 550W 80+ Bronze Certified ATX Power Supply ($29.99 @ Newegg)
Operating System: Microsoft - Windows 10 Home OEM 64-bit ($94.89 @ OutletPC)
Base Total: $667.72
Promo Discounts: -$10.00
Mail-in Rebates: -$55.00
Shipping: $0.99
Total: $603.71
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-09-06 13:28 EDT-0400
I did see a good deal on a refurb tower for $300, which will save you the $100 microsoft fee over BYO
Vostro i3-8100, 8GB Ram, 1TB HDD, win10 refurb $300
...and you just need the gpu/psu:
PCPartPicker part list:
https://pcpartpicker.com/list/gTZktJ
Price breakdown by merchant: https://pcpartpicker.com/list/gTZktJ/by_merchant/
Video Card: MSI - GeForce GTX 1050 Ti 4GB Video Card ($164.89 @ OutletPC)
Power Supply: Corsair - CX (2017) 550W 80+ Bronze Certified ATX Power Supply ($29.99 @ Newegg)
Base Total: $229.88
Mail-in Rebates: -$35.00
Total: $194.88
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-09-06 13:30 EDT-0400
So that saves you about $100 total with the budget pre-build and gpu/psu upgrades with the downside of case airflow and budget motherboard.