With the French midfielder Dimitri Payet [di.mi.tʁri.pa.jɛt] achieving national hero status for the host nation with his exploits in the ongoing UEFA Euro 2016 football tournament, a newspaper from his home island territory of Reunion published an article exclaiming in its headline, “Maintenant, ils savent que ça se prononce ‘paillettes’ [Now, they know that it’s pronounced ‘pie-ette’]”.
Payet is a common surname in this overseas department of France in the Indian Ocean, where it is pronounced [pa.jɛt], identical to the French word paillettes. In English, this could be approximated as PY-et [ˈpaɪ̯.ɛt] or py-ET [paɪ̯.ˈɛt] (different dialects of English have different preferences when it comes to stressing French names). However, it is fair to say that until now the continental French were mostly not familiar with this pronunciation. The article noted that even the comedian Manu Payet had to resign to being called Paillé, with a silent t.
It was only those in the know who were pronouncing the t, but crucially, these included the French football commentators. The millions of French viewers who tuned into the matches to witness Dimitri Payet exploding onto the scene were thus made aware of the unusual pronunciation of Payet. It is a small detail perhaps, but the author of the article certainly derived satisfaction from this.
Apart from in recent loans such as Najat [na.ʒat] from Arabic, word-final t following a vowel is nearly always silent in French names. You see this in Fermat [fɛʁ.ma], Mahaut [ma.o], Monet [mɔ.nɛ], Petit [pə.ti], and Peugeot [pø.ʒo], for example. It is no wonder that most French speakers not familiar with the name would pronounce Payet with a silent t.
But there are exceptions. Some are short, one-syllable names like Lot [lɔt], the name of a river in the southwest of France and the department named after it, and the surname Cot [kɔt]. Final t is also usually pronounced in more obscure Hebrew or Latin names, like the biblical name Josaphat [ʒo.za.fat] or Expédit [ɛks.pe.dit], the name of a Roman Catholic saint.
The name Moët [mɔ.ɛt, mwɛt], famous for the champagne, also is properly pronounced with the final t, as is the surname Abauzit [a.bo.zit]. The Canadian French surname Ouellet [wɛ.lɛt] is yet another example, and now the Reunionese surname Payet can be added to this list.
Most instances of word-final t in Old French ceased to be pronounced as it developed into Middle French and then Modern French. There were variations in which words kept the t sound, which is why even today there are cases such as août [u(t)] and but [by(t)] where the final t may be either silent or pronounced. In Canadian French, final t tends to be pronounced more frequently than in European Standard French, e.g. juillet is [ʒɥi.jɛt] instead of [ʒɥi.jɛ]; this is why the t is pronounced in the Canadian surname Ouellet [wɛ.lɛt]. In fact, Canadian French also extends this tendency to other surnames such as Brunet, pronouncing it [bʁy.nɛt] whereas European Standard French has [bʁy.nɛ] with a silent t. However, Ouellet is almost exclusively Canadian, which is why the pronunciation with the final t is recommendable even in Standard French. As Payet is almost exclusively Reunionese, it seems similarly reasonable to follow the local pronunciation with the final t even in Standard French, as is currently happening.
While most surnames ending in -et or -ot ended up with silent t in Standard French, it seems that a select few such as the Reunionese Payet and Canadian Ouellet ended up with the t pronounced because the more peripheral varieties of French spoken in the areas where they were used were more likely to retain the final t sound.