Juno
The Roman queen of the gods, for whom many said the month of June was named; her temple as Juno Moneta was dedicated on the first of June.
How the Romans fixed the year
Censorinus, On the Birthday 20–22 (238 CE), in the edition of Friedrich Hultsch (Leipzig, 1867) · 238 CE
Maium vero non a maioribus, sed a Maia nomen accepisse, quod eo mense tam Romae quam antea in Latio res divina Maiae fit et Mercurio. Iunium quoque a Iunone potius quam iunioribus, quod illo mense maxime Iunoni honores habentur.
May, he says, took its name not from the elders but from Maia, because in this month sacrifice is made to Maia and Mercury, now at Rome as before in Latium. June too came from Juno rather than from the younger men, because in that month Juno is honored most of all.
Where the months got their names
Macrobius, Saturnalia 1.12–14 (early fifth century CE), in the edition of Franz Eyssenhardt (Leipzig, 1893) · early fifth century CE; the dialogue is set in the 380s
Iunius Maium sequitur aut ex parte populi, ut supra diximus, nominatus aut, ut Cingius arbitratur, quod Iunonius apud Latinos ante uocitatus, diuque apud Aricinos Praenestinosque hac appelatione in fastos relatus sit adeo ut, sicut Nisus in commentariis fastorum dicit apud maiores quoque nostros haec appelatio mensis diu manserit, sed post detritis quibusdam litteris ex Iunonio Iunius dictus sit. nam et aedes Iunoni Monetae Kalend. Iuniis dedicata est.
June follows May. It was named either after a part of the people, as I said above, or, as Cingius judges, because the Latins once called it Iunonius [after Juno]. The people of Aricia and Praeneste long kept this name in their calendars. As Nisus says in his notes on the calendar, our ancestors too kept this name of the month for a long time, but later some letters wore away and Iunonius became Iunius. Indeed, the temple of Juno Moneta was dedicated on the first of June.