I wasn't able to find out why this word, or name, has a smooth breathing.
As you may have seen already, LSJ actually has separate entries for "Raros", a name, and "raros", the "embryo" word that TKR's answer deals with. It's unclear to me if these have the same origin.
LSJ seems to have a bit more information and examples of words related to the name in its entry for "Rarion pedion":
A. of Rarus, where tillage was first practised, and which was sacred to Demeter, Paus.1.38.6, St.Byz.; (without πεδίον) Ρ̓άριον, τό, h.Cer.450; Ρ̓αρία, ἡ (sc. γῆ), Plu.2.144b; whence the goddess was herself called Ρ̓αριάς, ἡ, St.Byz. [α is long, h.Cer. l.c., so that the accent is prob. not Ρ̓άρος, as in most codd.:—for the smooth breathing, v. Hdn.Gr.2.402, 940.]
Here is a relevant Quora post: Why is “ρ” translitered as “rh” instead of “r”? Philip Newton's answer cites a passage from Gaps in the System by
rho with smooth breathing appears [...] in Ancient representations of psilotic dialects. The TLG corpus manages 57 instances, all of them Aeolic, with the exception of Rarus, father of Triptolemus, the first human to have been taught agriculture by Demeter; his name was apparently back-formed from the (non-Attic) Rarian field in Eleusis associated with the cult of Demeter. In the surviving corpus, Rarus and his field are predictably mentioned more by the grammarians, who loved a good exception, than by the classics themselves:
Τὸ ρ ἀρχόμενον λέξεως δασύνεσθαι θέλει, ῥά, ῥανίς, ῥάξ χωρὶς τοῦ Ρ̓ᾶρος (ἔστι δὲ ὄνομα κύριον) καὶ χωρὶς τῶν ἐξ αὐτοῦ οἷον «Ρ̓αρίδος Δηοῦς» καὶ Ρ̓αρία γῆ καὶ Ρ̓αριάς, σημαίνει δὲ τὴν Δήμητρα, καὶ Ρ̓άριον πεδίον ἐν Ἐλευσῖνι, ἐπὶ τούτων γάρ φασι τὸ μὲν πρῶτον ρ ψιλοῦσθαι, τὸ δὲ δεύτερον δασύνεσθαι. ἀλλὰ τὰ δύο ρρ ψιλωτέον ἐπὶ τούτων καὶ κατὰ τὴν πρώτην συλλαβὴν καὶ κατὰ τὴν δευτέραν. (Herodian, De prosodia catholica 3.1.547)
At the beginning of a word, r takes a rough breathing, as in rhá "then", rhanís "drop", rháx "grape", except for Rarus (a proper name) and the words derived from it, such as "Rarian Demeter" and "Rarian earth" and Rarias, which is a name for Demeter, and the Rarian field in Eleusis. It is said that for these the first r has a smooth breathing and the second a rough; but the two r's in these words should bear a smooth breathing both in the first and in the second syllable.
Not all authorities put up with the exception; the Suda preferred to call Demeter Rharias.
I also found a short allusion to dialectal variation in the realization of rho in an extract from the book Asianic Elements in Greek Civilisation (1938), by William Ramsay, "Chapter XXI: General Principles", hosted by The Gifford Lectures online:
The letter “r” appears and disappears in perplexing fashion in Anatolian words as written in Greek alphabet. Probably a soft “r” was used in Asia Minor as in the English word pretty; but there were doubtless more sounds than one “rho” in Anatolian although they cannot now be distinguished. In Greek the “rho” was rougher and is written ῥ except in Aeolic (which is more Anatolian in type).
As far as I know, all modern dialects of Greek are psilotic; according to Wikipedia
The loss of /h/ happened at different times in different dialects of Greek. The eastern Ionic dialects, the Aeolic dialect of Lesbos, as well as the Doric dialects of Crete and Elis, were already psilotic at the beginning of their written record. In Attic, there was widespread variation in popular speech during the classical period, but the formal standard language retained /h/. This variation continued into the Hellenistic Koine. [...] By the late Roman and early Byzantine period, /h/ had been lost in all forms of the language.
So at later time periods at least, and in earlier time periods for some varieties of Greek, there would have been no difference in pronunciation between rho with a rough and smooth breathing. It would just have been a spelling exception to memorize.