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According to Wikipedia's Gilbreath's conjecture page,

The statement is named after mathematician Norman L. Gilbreath who, in 1958, presented it to the mathematical community after observing the pattern by chance while doing arithmetic on a napkin. In 1878, eighty years before Gilbreath's discovery, François Proth had, however, published the same observations along with an attempted proof, which was later shown to be false.

I did a few online searches regarding why Proth didn't get his name associated with this conjecture, but didn't find very much. With Bing, I did a search with

Why isn't Gilbreath's conjecture called Proth's conjecture

Among the results, in the Math Overflow answer Is there any progress toward solving Gilbreath's conjecture?, Gerry Myerson's comment has

The conjecture is problem A10 in Guy, Unsolved Problems In Number Theory. In 1878, long before Gilbreath made the conjecture (1958, unpublished), Proth claimed to have proved it - Guy gives the bibliographic details.

In the Reddit post Proth's incorrect proof of Gilbreath's conjecture., mikabast's post has

... while citations of Proth's failed proof abound in research papers it appears that the citation has taken the place of the actual paper. It has become a form of mathematical folklore, where many authors of papers on the Gilbreath conjecture cite Proth's proof without having actually seen it. For example, through personal communications with Andrew Odlyzko (February 5, 2012), he admitted not remembering whether he ever saw Proth's proof even though he cites it within his paper. Subsequent papers then cite Odlyzko's work on the conjecture and include Proth's proof even though it is highly unlikely that the author has seen it. Unfortunately, this paper continues that tradition.

The post also says the above text is from "from Kyle Surgill-Simon's recent (2012) thesis: http://www.carroll.edu/library/thesisArchive/Sturgill-Simon_2012final.pdf ".

I next did a Google search using

Proth Gilbreath conjecture

Among these results, the rxiv.org paper "An Elementary Proof of Gilbreath’s Conjecture" states

To begin the story, the anecdote goes that an undergraduate student named norman Gilbreath was doodling on a napkin one day in a cafe and found a very interesting characteristic of the list of sequential prime numbers and the diferences between them. He proposed that these diferences, when calculated repetitively and left as absolute values, would always result in a row of numbers beginning with 1 (after the first row). No one has been able to prove it. In 1878, eighty years before Gilbreath's discovery, François Proth had, however, published the same observations along with an attempted proof, which was later shown to be false.

I didn't look at the paper's so-called "elementary proof" very carefully, but from what I saw, I'm quite certain it's false. As such, I'm not overly confident of the quality of anything in this paper, including what I quoted above, but I'm providing it here for completeness.

Can anybody provide any specific reasons why this conjecture was named Gilbreath's conjecture rather than something like Proth's conjecture instead?

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    $\begingroup$ In math, as everywhere else, the naming of conjectures/results is often an accident of history. It correlates more not with who was "first", but with who made it well-known and convinced people it was important/interesting. Sometimes it is the same person. But if not, Gilbreath's case is typical, see e.g. Where did the story about Newcomb observing Benford’s Law come from? $\endgroup$ – Conifold Sep 6 at 4:45
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    $\begingroup$ The general situation is described by Stigler's law of eponymy. $\endgroup$ – Peter Taylor Sep 6 at 7:17
  • $\begingroup$ Are you aware of the fact that Pythagoras' theorem was known in Babylon 1000 years before Pythagoras was born? $\endgroup$ – José Carlos Santos Sep 6 at 8:20
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    $\begingroup$ The things are not always named after their first discoverers. Many people noticed that rather the opposite is true (see "Arnold's Principle", for example). Some general patterns of naming mathematical objects after people are discussed here: mathoverflow.net/questions/285627/… $\endgroup$ – Alexandre Eremenko Sep 6 at 17:28

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