Best $500 System?

I'd go with a NuForce DDA-120, an integrated amp with a nice DAC. It is an Absolute Sound Editors Choice at $499. Currently on sale at Audio Advisor for $299. Available in black or white. It comes equipped with a Bluetooth dongle. 
If your friend has a DVD or CD player with a digital coax out, buy a cable & hook it into the NuForce 's digital input.You avoid the cheap internal DAC in the player and now have a CD transport connected to a very capable DAC & great amp. If she doesn't have a player or it doesn't have a digital out, (optical or coax - the NuForce handles both), then  buy a DVD player with a digital coax or optical out. Best Buy, Target, and other discounters have a variety for $35-50. I'd stick with a major brand, like Sony, Samsung, Phillips, or LG. The DVD players work, but are usually slower to cue up than a CD player. On the speaker front I'd go with Pioneer SP-BS22-LRs. Best Buy usually has them at list $129 pair, but periodically takes them down to $100.  I'd try Audio Video Logic on line, they have them on special at $99 a pair. Not sure if they would ship them for free, but they are excellent retailers/enthusiasts -so talk to them. Usually Best Buy will match a bonafide authorized retailers online price, but if it is a wash give the specialty store your business and start a relationship. I've owned the Pioneers (as well as the other Andrew Jones darlings the ELAC- B5, B6 & UB 5) I think the NuForce amp and Pioneer speakers would have great synergy. Should be a nice system for CDs and Bluetooth streaming.

Bet those Reference 20s sound good!
Too bad there isn't a digital audio out of your CD player. Often a digital coax output is nestled among the component connections. The nice thing about the NuForce/Pioneer/DVD/CD system is that it's all new, with current technology and warranties. 
If you do need to go the DVD player route, there are a wide variety of prices & capabilities. At the upper end,($45-80), the DVD players have higher video resolution and streaming capability (if she wants to take advantage of the video capabilities),and still be below $500 total.
FYI (for down the road)-Last Christmas season, authorized dealers were running the compact Pioneer SW-8 MK2 subwoofer for $99 on Amazon. It's  a great match with the Pioneer bookshelf system. From your first post, it doesn't sound like she is focused on bass performance, but when you add the sub, tuned to a barely perceptible level, the combo becomes ridiculously good from top to bottom. In addition to a line level subwoofer out, the NuForce has an analog input so you can run a turntable at a later date , (with an outboard phono preamp). You can also use your CD connected to the analog inputs, without running it through the NuForce's DAC if you are really trying to save money.
Parts Express has a 24" steel stand for $55 pair that work well with the Pioneers.

Bet those Reference 20s sound good!
Too bad there isn't a digital audio out of your CD player. Often a digital coax output is nestled among the component connections. The nice thing about the NuForce/Pioneer/DVD/CD system is that it's all new, with current technology and warranties. 
If you do need to go the DVD player route, there are a wide variety of prices & capabilities. At the upper end,($45-80), the DVD players have higher video resolution and streaming capability (if she wants to take advantage of the video capabilities),and still be below $500 total.
FYI (for down the road)-Last Christmas season, authorized dealers were running the compact Pioneer SW-8 MK2 subwoofer for $99 on Amazon. It's  a great match with the Pioneer bookshelf system. From your first post, it doesn't sound like she is focused on bass performance, but when you add the sub, tuned to a barely perceptible level, the combo becomes ridiculously good from top to bottom. In addition to a line level subwoofer out, the NuForce has an analog input so you can run a turntable at a later date , (with an outboard phono preamp). You can also use your CD connected to the analog inputs, without running it through the NuForce's DAC if you are really trying to save money.
Parts Express has a 24" steel stand for $55 pair that work well with the Pioneers.

Sorry about the duplicate entry above.
To get a better idea of how good the NuForce DDA-120 is, check out the rave Sound Stage review when it first came out in 2015, at a list price of $699!  (The Absolute Sound recommended it then too at $699, and continues to do so in 2017)
It seems to me that it was NADs de facto lowering of the list price on the D-3020 from $499 to the widely available street price of $399, that started a big price correction in this category (PS Sprout, etc)
NuForce  cut the list for the DDA -120 from $699 to $499. 
The sale at $299 is just crazy good, great time to be a budget audiophile!

 I found the NuForce to be very crisp and open sounding with decent bass. The B-6s are sort of like Pioneers with improved extension at both ends of the spectrum and smoother highs. I enjoyed both combinations (NuForce/Pioneer & NuForce/ELAC). I’m not sure what is going on with NuForce as a company. I get the impression, from their odd website, that they aren’t actively involved in the audio side (other than earphones)- at least I can’t get any other products to pop up. You’ve probably read that Jason Lim, a principle at NuForce, split off to form Nuprime which is producing a number of well interviewed audio products, ( like the IDA 8 & IDA 16 integrated amps). I wonder if the lower prices on NuForce DDA-120 indicates they are discontinued. If true, the good news is you get this quality of sound & industrial design for $249, the bad news may be that product support may be lacking at some point. I would buy from Audio Advisor, or Amazon that will be responsive if your aren’t happy. Not sure how soon you are looking to buy the ELACs, but Music Direct runs some nice sales, don’t remember what they had the B-6s for, but the Unifi UB-5s were $349, around Christmas.

My last post may have seemed down on NuForce-please don’t take it that way, its  a great amp,(at a great price)- with a family resemblance, sound wise, to the Nuprime IDA 8 (IMO).
Just wondering about the company commitment to audio, especially two channel.
FYI-I picked up the UB-5s from Music Direct at the end of 2016 for $399 pair, but this Christmas pricing was an even better deal.