Frommer's "Easy Guides" series are really good. Lightweight so easy to tote around in a day bag or purse, cover a decent selection of sights, clear info on best ways to get around using public transit, etc. If your mom has a smartphone, there's pretty much an app for just about everything she'd want to see in NYC, so I would not go overboard on the guidebooks. Make sure she downloads a good app that she can use to get around on the MTA.
As you go there often, you may want to invest in the "Not for Tourists: NYC" guide for yourself and lend that to her in addition to getting her the Frommer's Easy Guide on NYC. While NFT does provide some very limited info on museums, galleries, etc., it's great as a companion guide to reference when you need to look at a neighborhood map, find a place to eat or a local bar.
For ETTBD, if you have the 2013 edition, I would not see the point purchasing an updated edition. I feel like that particular book is geared more towards those who have never traveled in Europe at all, or who haven't traveled there for the last decade or more. It's a lot of travel skills compiled into a guide, and most of it is covered by watching one of the Rick Steves' travel skills videos (which you can probably view here, and if you can't, you can find it on YouTube somewhere).