My great grandfather was an immigrant from Ireland who left as a child. He died in the 1970s. My general understanding of things via my grandmother (who knew him well and was in her 30s when he died) has been that he immigrated with his family, his parents took sick and died soon after immigration to New York when he was a child, and that he was adopted by an American family. I've always understood it that he was born in the 1870s, but I have done some digging and found a few of his census records from 1920, 1930, and 1940. Most of his information is consistent (first language written as Irish, years working, etc). However, he gives different years of birth on each of them, with a wide variance of over a decade. He also gives consistently different years of immigration to the US as well as his naturalization as an American citizen. The latest one he gave was 1900 for immigration. For the 1920 census, he even wrote 1919 as his year of naturalization as a citizen, a full decade or two later than I would have assumed.
What would possess someone to give such widely divergent information on the census? He was a lifelong manual laborer, so I have heard that possibly lying to look younger for work reasons may have been it, but that still seems like a bit of a stretch to me.
After rereading I've gone ahead and approved this. If you don't get an answer over the next few days, you might want to reconsider reposting with more details in the title. As is, the title is unlikely to draw many experts and looks very much like an example seeking question.
It's possible that your great grandfather wasn't lying at all, but that he was at work and someone else answered the census worker's questions for him. Poor Irish immigrants often lived in large, crowded apartment buildings. Sometimes the census worker would catch people from the family itself, but if they were out, a neighbour might have to answer for them. Neighbours might have only a vague idea of a person's age and how long they had been in the country. Even family members might not know exactly; for example, if a young woman were asked to answer how old her grandfather was or how long it had been since he left the "old country", she might have only a guess. Different people would have answered at every census, which causes further discrepancies.
Immigration and naturalization dates would not have been very close together, so a difference of 19 years would have been pretty common. This is because in order to be naturalized, you had sign a document declaring that you had been living in the United States for a certain period of time (or sometimes other requirements, such as arriving underage but now being an adult; this varied across time and place). For example, one of my own ancestors came to New York from Ireland in 1855 but was not naturalized until 1872. Historically, naturalization sometimes required classes or training, another reason it was rarely immediate.
If you'd like to learn more about individual censuses and how they were conducted, which differed each year and also between state and federal censuses, Familysearch.org has a lot of resources and is free to use.
You might want to hit r/genealogy for this as well