The information about Loic in this site is not correct.
Loïc is a Provençal and Languadocian ancient form of Louis (pronounced loh-EEK) documentated since 12th century, in the ancient spelling Loic, (with other variants as Lois, Loys, Lozoic… nowadays Loís, Lozoïc).
In 19th century, the Breton poet Aogust Brizeug (Auguste Brizeux in French) (1803-1858) published Mari (1828), a book with a poem named “Loic’s song”, which was quickly popularised –also because Berlioz’s and Massé’s lieds-.
In normative Breton the name is spelled Loig (in 19th century documents it can be found as Loic), but it is possible to find it spelled Loïc in French, with a dieresis over the I to break the diphthong.
People usually think that Loig (pronounced loh-EEK) is a Breton nickname, nowadays used as full name, because of the ending in –ig (-ic, -ick in other spellings), which is a usual ending of Breton nicknames; and they think that it was originally a nickname of Louis.
In fact, in Breton Louis is Loiz (nicknames Loeizig and Loizig) and Loig seems, according the Breton philologist Gwennole Le Menn, a misreading of Laouig, a nickname of Gwilherm, the Breton form of William.
So Loig as Breton name is not previous to the middle of the 19th century, but a hundred of years later it was very popular among Bretons. The problem to find Loig/Loic in French documents in 19th and 20th centuries is that the French naming law before 1966 forbade not-French names (that is Breton names, Catalan names, Basque names, Occitan names…) and Breton Loigs appear under the name Louis.
Since ‘80s and especially ‘90s Breton names were a trend among French people. Nowadays they are very popular, but they have fallen a bit in popularity in favour of Italian and Spanish names (Leo, Esteban).
In Loïc there is another convergent path. In French there is the name Eloïc, probably from the Germanic personal name Haluig, from which Loïc can be a nickname (or simply it can be related in terms of popularity).
The etymology of Haluig is not clear. It could come from helm, “helm”, and wig, “fight, battle”, or from heil, “happy” (from heilig, “saint”), and wig, “fight, battle”, with attraction of Eloi (in fact, Eloïc is often misinterpreted as diminutive of Eloi).