I researched this a couple of year's ago for my elderly dad. He had a landline that has unlimited long distance so his mobile phone just needed to provide emergency service for when he is driving or traveling. I found the perfect solution for him at Walmart by pre-paying a couple of hundred bucks for a bundle of minutes good for a year and a free flip phone. He dislikes WallyWorld so he found a similar deal at a local Radio Shack (now closed), albeit a little more expensive.
Do not let them get suckered in on a monthly reload deal. Sure, the minutes roll over but I have another elderly client that now has a zillion minutes and still is paying a monthly service fee. She went thru Radio Shack also and I guess she was comfortable with the cost for the peace of mind.