I am assuming that his old computer is not running Windows 8/Windows 8.1, so I would avoid buying a computer with either of those OS's. I am 35 years old and a self-described computer nerd, but I still had a not so pleasant time adapting from Windows 7 to Windows 8. You will probably want to go with something that he is already familiar with to keep it from being a headache for him. With that said, here is a list of some decent laptops with at least a 16" screen and under $600.00 per your request. All computers that I listed also run Windows 7.
http/www.dell.com/us/p/inspiron-17-3737/pd?oc=fcncwd1224sw7&model_id=inspiron-17-3737
http
/www.amazon.com/Acer-Aspire-17-3-Inch-Windows-Premium/dp/B00CLZWK64/ref=sr_1_3?s=electronics&ie=UTF8&qid=1394144060&sr=1-3&keywords=Laptop+windows+7
http/www.amazon.com/Dell-Inspiron-i15RV-953BLK-16-Inch-Textured/dp/B00F4MEK6E/ref=sr_1_1?s=electronics&ie=UTF8&qid=1394144060&sr=1-1&keywords=Laptop+windows+7
http
/www.amazon.com/HP-Pavilion-17-e040us-generation-Processor/dp/B00EKVXAAC/ref=sr_1_2?s=electronics&ie=UTF8&qid=1394144060&sr=1-2&keywords=Laptop+windows+7
http/www.amazon.com/Acer-V3-731-4439-17-Inch-Pentium-Processor/dp/B00F0RD86G/ref=sr_1_4?s=electronics&ie=UTF8&qid=1394144060&sr=1-4&keywords=Laptop+windows+7
http
/www.amazon.com/Acer-Gateway-17-Inch-E1-1200-storage/dp/B00FSVK0IS/ref=sr_1_6?s=electronics&ie=UTF8&qid=1394144060&sr=1-6&keywords=Laptop+windows+7
http/www.amazon.com/Toshiba-C55-A5281-Processor-SuperMulti-Windows/dp/B00DXWFF3E/ref=sr_1_5?s=electronics&ie=UTF8&qid=1394144060&sr=1-5&keywords=Laptop+windows+7
http
/www.amazon.com/Acer-P2020M-NX-MGAAA-004-E1-731-4699-17-3-Inch/dp/B00I3DOZEI/ref=sr_1_7?s=electronics&ie=UTF8&qid=1394144060&sr=1-7&keywords=Laptop+windows+7
The prices of these range from less than $400.00 up to but never exceeding $600.00. I included computers of many brands, as you or him may have a favorite that I am unaware of. Me personally.....I have always liked Dell and Acer laptops. I also chose only computers that had a good customer rating. Consumers have nothing to gain or lose when reviewing a product, so they typically tell the truth!
Good luck with finding a good computer for grandpa! I'd give anything to still have my grandpa. I lost him in December. He was my best friend and I miss him greatly.