Last week I read a notice about the New York Times opening up their archives. They are every name search able from 1851 to the present, but some you can not view online due to the copyright. (Note: The Spokane downtown library has the ones not online on microfilm). While you probably will not find your ancestor in the Times, you can search for the ship they came on. I searched for my grandfather's ship, the SS Nevada, and about 20 hits came up, several close to when he came to the USA. Another tip is to search for your ancestors hometown. I searched for my grandfather's town in Minnesota (Wadena), and got a few hits. Searching for Spokane brought up coverage of the Spokane Fire in 1889.
When on the Times homepage, near the middle is a drop-down menu with search times; click there and change to 1851-1980 and then type in the name you are looking for. Here is the URL for the New York Times: http://www.nytimes.com/