I'm trying to figure out what 8 U.S. Code § 1401 said in 1961 in order to resolve an argument on the Internet.
I've found changes to the law like Public Law No: 103-416 but not the full text at a particular time.
The Immigration and Nationality Act dates from 1952, meaning that in 1952 Congress enacted an entirely new codification of federal immigration law. This law was not phrased as amendments to the old laws; it completely replaced them and was a self-contained law.
With this section, you're in luck. In 1961, §301 of the INA (aka 8 USC §1401) had not yet been amended. If you look at the bottom of the LII page you linked, you can see all the laws that modified this section of the US Code and their years, and the first one after 1952 was in 1966. So, you want §301 of the INA as enacted. LII reports that §301 appeared in volume 66 of the US Statutes at Large (which lists things exactly as they were passed by Congress), at page 235. You can find that here.
That's the law as it would have read if you looked it up in 1961. However, changes to a law can sometimes be retroactive. For instance, a 1966 change said that people working for the US government or certain international organizations, as well as their unmarried dependent children, count as being in the US for residency requirements. This provision also said that it applied to people born after 1952 as though it was part of the original INA as enacted. It was not, though, actually part of US law in 1952. If someone was born in 1952 in Germany to a noncitizen parent and a citizen parent, where the citizen parent was the child of a US soldier and had spend most of their childhood living with their parent on US military bases abroad, then that child would not have been considered a citizen in 1952. When the law changed in 1966, they would be considered a citizen from birth, but would not have attained that status until the age of 14.