The basic premise in family history is to start with what you know, and work backwards and outwards from that point in small increments to discover more. You want to be able to say with confidence, when you recognize a record associated with your relative, that you are looking at something that was recorded about your relative and not someone else with the same name.
One approach to finding more records might be to write a biographical profile and a timeline for your 2-great-grandfather's life. Make a note of all you think you know, and pay attention to where and how you learned it. Then look for other records that might confirm those bits of information. If your 2-great-grandfather had siblings, look at their records, too.
Of particular interest are any naturalization papers that may be associated with your family. As a part of the naturalization process, immigrants had to give the date and port of arrival for their entry into the USA. This in turn can be a pointer to the passenger manifest, which can contain other clues like the immigrant's home town.
There is a persistent myth that immigrants' names were changed by the inspectors at Ellis Island. This was not the case, because passenger lists were filled out at the point of origin, not in the US. Many immigrants did change their name during the immigration process, by choice. See these articles for more details:
- Why Your Family Name Was Not Changed at Ellis Island (and One That Was) by Philip Sutton, Milstein Division of United States History, Local History and Genealogy, Stephen A. Schwarzman Building, posted July 2, 2013 by the New York Public Library
- The myth that refuses to die: “Ellis Island changed our name” by Aliza Giammatteo, posted December 13, 2012; originally published in the National Italian American Foundation’s (NIAF’s) Ambassador Magazine
- American Names: Declaring Independence by Marian L. Smith, senior historian for the U.S. Immigration and Naturalization Service, Washington, D.C.
Unfortunately big-box genealogy vendors like Ancestry.com make it seem simple -- all you have to do is fill in a search form with a name and records cascade out like magic. But it isn't always that simple. Finding records requires an understanding of what records were created, and why they were created; the more you understand about when and why, the better your search results will be. If a straightforward search for "Stanislaw Woytinkewicz" is not yielding any results, then you have to search another way. It might help to broaden your search, and to narrow your search, depending on what collections you are searching.
Keep a log of what record collections you looked at, why you thought a record about your 2-great-grandfather might have been in that collection, what search term you used, and what your results were, especially if you didn't find anything.
If you edit your question to add more information about your 2-great-grandfather's life and when his family entered the US, it will be easier for people to give you specific advice.
Also consider that your 2-great-grandfather had family, associates, and neighbors, and looking for other people associated with him might yield some clues. See this article by Elizabeth Shown Mills, QuickLesson 11: Identity Problems & the FAN Principle.