Fleursoup

For premiere there is a extention www.voukoder.org That allows you to gpu exporting on nvidia and amd gpus for much faster exporting, and let you use x264 and x265 which are better encoders that what is included in premiere(so better quality for same time, or faster for same quality)

For this I'd definitely go for a Ryzen CPU
PCPartPicker part list / Price breakdown by merchant Type Item Price CPU AMD - Ryzen 7 2700X 3.7 GHz 8-Core Processor $308.99 @ Walmart Motherboard ASRock - Fatal1ty B450 GAMING K4 ATX AM4 Motherboard $102.98 @ Newegg Memory Corsair - Vengeance LPX 16 GB (2 x 8 GB) DDR4-3000 Memory $129.99 @ Amazon Storage Samsung - 970 Evo 250 GB M.2-2280 Solid State Drive $77.99 @ Amazon Storage Seagate - Barracuda 2 TB 3.5" 7200RPM Internal Hard Drive $59.89 @ OutletPC Video Card Gigabyte - Radeon RX 580 8 GB Gaming 8G Video Card $288.00 @ OutletPC Case NZXT - H500 (Black) ATX Mid Tower Case $69.99 @ Amazon Power Supply Corsair - RMx (2018) 750 W 80+ Gold Certified Fully-Modular ATX Power Supply $99.99 @ Amazon   Prices include shipping, taxes, rebates, and discounts     Total (before mail-in rebates) $1197.82   Mail-in rebates -$60.00   Total $1137.82   Generated by PCPartPicker 2019-01-03 16:05 EST-0500   Something like that should work. Even without the mail-in rebate it's just under your budget.

Welcome to the forum!
 
I'd definitely build new, those are both rather dead ends.
This is a touch over budget, but once you sell your old hardware it should be right around $950.
 
PCPartPicker part list / Price breakdown by merchant
CPU: AMD - Ryzen 5 1600 3.2GHz 6-Core Processor  ($197.28 @ OutletPC)
CPU Cooler: CRYORIG - H7 49.0 CFM CPU Cooler  ($29.89 @ OutletPC)
Motherboard: MSI - B350 TOMAHAWK PLUS ATX AM4 Motherboard  ($91.98 @ Newegg)
Memory: G.Skill - Ripjaws V Series 16GB (2 x 8GB) DDR4-3200 Memory  ($179.99 @ Newegg)
Video Card: Zotac - GeForce GTX 1070 Ti 8GB AMP Edition Video Card  ($469.99 @ Amazon)
Power Supply: Corsair - CXM 550W 80+ Bronze Certified Semi-Modular ATX Power Supply  ($49.99 @ Amazon)
Total: $1019.12

Assuming you'll use rebates, and are willing to go over budget a bit. For just 1080p, a 1060 6GB or RX 580 is fine, that'll cost $200 less. The 1070 Ti will perform quite a bit better than the 1060. 
PCPartPicker part list / Price breakdown by merchant
CPU: AMD - Ryzen 5 1600 3.2GHz 6-Core Processor  ($197.28 @ OutletPC) 
Motherboard: MSI - X370 GAMING PLUS ATX AM4 Motherboard  ($80.98 @ Newegg) 
Memory: G.Skill - Aegis 8GB (1 x 8GB) DDR4-3000 Memory  ($87.99 @ Newegg) 
Storage: Crucial - BX300 240GB 2.5" Solid State Drive  ($74.99 @ Amazon) 
Storage: *Western Digital - RE4 1TB 3.5" 7200RPM Internal Hard Drive  ($42.93 @ Amazon) 
Video Card: MSI - GeForce GTX 1070 Ti 8GB Titanium Video Card  ($469.99 @ Amazon) 
Case: NZXT - Phantom 410 (Black) ATX Mid Tower Case  (Purchased For $0.00) 
Power Supply: Corsair - TXM Gold 550W 80+ Gold Certified Semi-Modular ATX Power Supply  ($49.99 @ Newegg) 
Total: $1004.15
Prices include shipping, taxes, and discounts when available
*Lowest price parts chosen from parametric criteria
Generated by PCPartPicker 2017-12-28 18:13 EST-0500

PCPartPicker part list: https://pcpartpicker.com/list/t8PfKZ
Price breakdown by merchant: https://pcpartpicker.com/list/t8PfKZ/by_merchant/
CPU: AMD - Ryzen 5 1600 3.2GHz 6-Core Processor  ($201.98 @ Newegg) 
Motherboard: ASRock - AB350 Pro4 ATX AM4 Motherboard  ($93.98 @ Newegg) 
Memory: Team - Vulcan 16GB (2 x 8GB) DDR4-2400 Memory  ($171.99 @ Newegg) 
Video Card: EVGA - GeForce GTX 1070 8GB SC Gaming ACX 3.0 Video Card  ($474.98 @ Newegg) 
Power Supply: Corsair - CX (2017) 650W 80+ Bronze Certified ATX Power Supply  ($49.99 @ Newegg) 
Total: $992.92
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-12-28 18:16 EST-0500

Do they need this much more storage? They already have it covered.

Ah, missed that part. If he doesn't have an SSD he can keep it, or he can just reduce the cost. 

I'd like to take a hundred. How can I geth it?