If you do decide on Mythology by Edith Hamilton, be sure to buy the 75th Anniversary Edition. Beautiful book, worth it for the illustrations alone :)
Could it be Edith Hamilton's Mythology? It's the first book that sprung to mind that's as comprehensive as you've described and would have been around in the 80s and 90s - a quick google search says that there have been green covers in the past, though maybe not matching your description entirely!
I fell in love with her writing the second I opened Shiver, so I make sure to buy all her books.
This is my mythology book:
Mythology: Timeless Tales of Gods and Heroes, 75th Anniversary Illustrated Edition https://www.amazon.co.uk/dp/0316438529/ref=cm_sw_r_cp_api_glt_fabc_BY82WJS5GA31YE05QYPW
The illustrations are gorgeous 😊
https://www.amazon.com/Mythology-Timeless-Heroes-Anniversary-Illustrated/dp/0316438529/ref=sr_1_2?ie=UTF8&qid=1534795752&sr=8-2&keywords=Mythology%3A+Timeless+Tales+of+Gods+and+Heroes This book has some seriously beautiful illustrations and the cover looks something magical for sure. And it might be a good way to introduce her some more 'serious' literature and bigger books. Oh, and btw, why not show her some games, like age of mythology? My team is developing a mythological game right now and it's all because we love Greek mythology and the AoM series since teenagers. =) (btw Persephone is my fav too haha)
I tried to stick to single-volume comics and illustrated mythologies since those should be quick to read and fun to look at
Also, I know you have a no-phones policy, but what if you encouraged the kids to read webcomics or ebooks on a tablet or laptop if you're able? I know a lot of book haters who are obsessed with webcomics because of the visual storytelling and ease of access so you should be encouraging non-readers to look into those (or to think critically about the TV and game stories they consume!)
Here's some printed volumes of webcomics:
I hope this helps!
I also just barely canceled my 3rd book pre order. Instead I got Mythology: Timeless Tales of Gods and Heroes by Edith Hamilton
It is actually pretty cheap on Amazon right now!
https://www.amazon.com/Mythology-Timeless-Heroes-Anniversary-Illustrated/dp/0316438529
this is the gold standard.
Mythology: Timeless Tales of Gods and Heroes by Edith Hamilton
https://www.amazon.com/dp/0316438529/ref=cm_sw_r_cp_apa_i_IPaKFbCK0NQ27