Cheap computer build

I don't think that for $350 you'll come anywhere close to what you already have. (Looks like your existing system is pretty nice. Certainly way better than mine.) If you're looking to build a computer for someone else, I would recommend buying a used computer on eBay rather than individual parts. I suspect that buying piece by piece (RAM, power supply, motherboard, CPU, CPU Fan, GPU, etc.) will be much more expensive than buying everything together. An i7 CPU by itself is more than $300 if purchased new and around $200 if purchased used so don't expect much for $350 - even if you buy used.
 
I'm using a computer with an older Core 2 Duo 3.0 GHz CPU and it seems to work fine. You can buy used computers with that CPU for much less than a newer i3/i5/i7 CPU. I haven't got any performance complaints, but my songs don't use a ton of plug-in synths.

I am not looking to match my current system.  just something that will give me decent performance for tracking
 

You are going to really struggle to get something under $350.  Power supply and case will run you at least $100 on their own (if you are lucky.)  The processor and mb are likely going to be almost $200 (and that's for an i3.)  That only leaves $50 for memory, harddrive and DVD.

He needs to get the mb/processor combo down to near $100 to have any shot if he needs everything else except an SSD.

I have built several PC's in the last few years using a case at about $40.00 and a PSU (400 W) at about $20.00 which were new items purchased on sale/rebate at Newegg.com. You can easily spend more than $150.00 for a case alone if you are persnickety. It should not be hard to find a new 1 TB HD for $40.00-45.00. Do not even think about getting an SSD on this budget. So long as you do not plan to overclock, stock coolers and standard desktop memory should be fine. My final builds typically came out at about $500.00 but the big cost was the MB and Intel chip, buying the models that were getting rave reviews a couple of years previously at substantial savings is the key. The law of diminishing returns applies bigly in computer performance, and a surprising number of top performing machines never get used in a way that shows the marginal performance benefits extra thousands will yield. The best way to buy your parts is to wait until what you need shows up at prices you can afford, and of course you will be spending a lot of time shopping that could be spent better doing almost anything else. A part time job to earn the money to buy a good pre-built might make more sense. 

I built a new system a while back with this case, power supply, and drives = $142.79, not counting CPU, RAM, mobo, and SSD.  That leaves $207.21 in your budget for the rest.  Find a good microATX board, CPU, and RAM for that and you're all set for $350.00 total
 
Rosewill Dual Fans MicroATX Mini Tower Computer Case FBM-01  $24.35
https://www.amazon.com/gp/product/B005LIDU5S/ref=oh_aui_detailpage_o07_s00?ie=UTF8&psc=1
 
Corsair CX Series, CX430, 430 Watt (430W) Power Supply, 80+ Bronze Certified  $50.66
https://www.amazon.com/gp..._s01?ie=UTF8&psc=1
 
Lite-On Super AllWrite 24X SATA DVD+/-RW Dual Layer Drive - Bulk - IHAS124-04 (Black)  $17.79
https://www.amazon.com/gp/product/B002YIG9AQ/ref=oh_aui_detailpage_o07_s01?ie=UTF8&psc=1
 
WD Blue 1TB SATA 6 Gb/s 7200 RPM 64MB Cache 3.5 Inch Desktop Hard Drive (WD10EZEX)  $49.99
https://www.amazon.com/gp...es01?ie=UTF8&psc=1