I'm sure there are other people here that would like to read news/websites when offline, so here is what I came up with
The NYTimes app can download the articles so it can be viewed offline, it just takes some time to download the articles.
There are two apps that allow you to view websites offline, Read it Later ($2.99) and Instapaper($4.99). Both offer a free app that limits the sites you can store and a "Pro" version. I haven't tested either one yet, but Read it Later seems to be more popular.
There is also a fluent news app that is free and is like the NYTimes app except that it downloads news articles from a variety of news sources. I just downloaded it and haven't had a chance to use it.
My question is if this saves all the articles on the iPhone, does it delete older articles, like if they are older than a day or do they just pile up?