There is only one piece of definite evidence that bears on the foundation of Min-Rimmon's beacon, in Unfinished Tales of Númenor and Middle-earth. There (I 2 (iii) Cirion and Eorl, to be precise) we are told that the three oldest beacons (Amon Dîn, Eilenach and Min-Rimmon) were built to communicate with the northern defenders of Gondor before the foundation of Rohan in III 2510. The suggestion there is that the first beacons were constructed because of the loss of the palantíri, which took place between III 1437 (when the Stone of Osgiliath was lost during the Kin-strife) and III 2002 (with the loss of Minas Ithil to the Nazgûl, and with it the Ithil-stone). Orthanc'spalantír also fell out of use during this period, probably with Gondor's abandonment of Orthanc in (probably) the seventeenth century of the Third Age. Min-Rimmon's beacon, then, was likely put in place during the latter half of the second millennium of the Third Age.
The meaning of Rimmon is uncertain. In his unfinished index to The Lord of the Rings Tolkien described it as a 'group of crags', but whether this was meant as an interpretation of the name, or merely a description, is not clear from the context. The element rim- can be read as a 'great number', so 'group of crags' is not implausible as an explanation of the name.