This is known as the Humanist Latin Orthography. The following is from a dissertation, "Accent Notation in the Classical Languages and its Influence on Lithuanian Accent Notation" by Mindaugas Strockis:
The archaic genitive singular -ās of the ā-stems (the first declension) is marked with a circumflex: aurâs, Majâs, paterfamiliâs, materfamiliâs, in imitation of Greek gen. sg. -ᾶς.
The genitive singular of the u-stems (the fourth declension) is marked with a circumflex, such as domûs, as if in imitation of the Greek εὐτυχοῦς, Σαπφοῦς, αἰδοῦς or similar.
The nominative singular -ūs of the Greek names ending in -οῦς in the original, is also written with a circumflex: Trapezûs, Hierichûs.
The genitive plural ending -um (= -ōrum) was written with a circumflex mark, such as deûm (versus unmarked acc. sg. deum), in imitation of the Greek gen. pl. ending -ῶν.
The genitive partitive of the personal pronouns nos, vos was marked with circumflex: nostrûm, vestrûm (versus unmarked acc. sg. masc. and nom./acc. sg. neuter of the possessive nostrum, vestrum), in imitation of Greek ἡµῶν, ὑµῶν.
The ablative singular of the ā-stems (the first declension) was marked with a circumflex: linguâ, Româ, versus unmarked nom. sg. lingua, Roma. This particular case is likely to have been indirectly influenced (or at least a posteriori justified) by Quintilian’s teaching (1.7.3) about the usage of the apex mark.
Sometimes the explicative diacritic of the humanist Latin orthography coincided with the actual word accent: nostrâs, Samnîs, which, according to Roman grammarians, could have been accented on the last syllable.
The grave mark on the last syllable was often used to distinguish adverbs, prepositions, and some other, usually indeclinable (Steenbakkers 1994, Burkard 2003), words: maximè (adverb) versus maxime (vocative), cùm (conjunction) versus cum (preposition), verò (conjunction) versus vero (dat./abl. sg.), quòd (conjunction) versus quod (pronoun) etc. For the logic of such orthography, compare also Quintilian’s mention (1.5.26) of the oxytonesis of auxiliary words, and the Greek oxytonic grave, especially frequent in prepositions (ἀνὰ, κατὰ, µετὰ, ὑπὸ etc).
Various humanist grammarians proposed also other orthographic rules to distinguish homographs, such as to mark the accent on the penultimate syllable according to the Greek third mora rule, such as to write lăbor and lābor respectively lábor and lâbor, or to write pône (imperative of a verb) but ponè (preposition).
Sometimes contracted or syncopated word forms were written with a circumflex (dî = dii, nîl = nihil, mî = mihi, nôrat = noverat, etc.) in imitation of the Greek contracted forms with the circumflex from the first mora accent: νόος > νοῦς, φάος > φῶς etc.
The system of explicative diacritics in Latin orthography was abandoned between 18th and early 19th centuries.