I could spend hours reading old, historical newspapers, which is funny because I don't tend to read current newspapers (between the internet and TV, I don't feel the need). At first, I would search them for the names of my ancestors, occasionally pausing to be amused by some of the advertisements. But once I had pretty much exhausted that, I started looking up major and local events which, although may have had no direct impact on my ancestor's lives, would have certainly been events which they read about in the very same papers. The idea of reading the same articles my ancestors would have was kind of cool to me, even if it didn't really help with my research, it does help me better understand their world.
I mostly read the Ambler Gazette, available from Access Pennsylvania Digital Repository, because some of my ancestors were in and around this area (and because the major Philadelphia newspapers don't have an online historic archive). It's really easy to lose myself in reading articles and learning more about the world my ancestors lived in and history of the areas I partially grew up in as well.
One of the more interesting articles I found was one about some local victims of the sinking of the Titanic. I was recently reading a novel that involved the Titanic (The House of Velvet and Glass - I highly recommend it, btw) and it inspired me to look up contemporary newspaper articles on it. In the Ambler Gazette, I found a tragic article about a couple in North Wales, Pennsylvania who received a message from their daughter-in-law informing them that their son (her husband), Austin VanBilliard, and his two children were aboard the Titanic. The couple had been unaware of this as Austin had wanted to surprise them with an early visit. At the time of the article, it was unknown whether Austin and his children survived or not. Upon looking at a Titanic victim's list, I discovered that sadly, none of them survived and I also noticed that the recovered remains had been buried in the same cemetery in Flourtown that some of my own ancestors are buried in. While it's doubtful that my ancestors knew this family, it still hits home how real history is.
Here's a clipping of the article that caught my interest:
But sometimes, the things that take my interest are more mundane. When I know the names of the churches my ancestors attended, it can be useful to look them up in newspapers and see what events or activities were going on in my ancestor's church life. It can also give me an idea of when significant local organizations or buildings were founded. Reading about this stuff is like watching your home town evolve in real time, which is not a feeling you get from reading a history of a location.
So don't overlook or dismiss newspapers just because you don't find (or have already found) a mention of one of your ancestors. Explore their world and get lost in it.