As user2922582’s answer says, use of the spelling “myrrh” in English is related to the use of the spelling “myrrhe” in French. The “rrh” in both of these goes back to the Latin form “myrrha”, which is a transliteration of Greek “μύρρα”.
The H occurs in the transliteration because of the double R. The general pattern for spelling /r/ in words taken from Greek is “rh” at the start of words, “r” in consonant clusters, and either “r” (for single “ρ”) or “rrh” (for double “ρρ”) between vowels (or word-finally after a vowel because of the loss of a final vowel in an Anglicized form, as in this case). “Mary” and “Maria” don’t have an H after the R because they have only a single R that isn’t the first letter in either of the names.
The Greek “rough breathing” corresponds to H
It’s true that the standard form of the Greek alphabet1 does not contain a letter corresponding to H, but Greek written with polytonic orthography has a number of diacritics. The use of H in the transliteration of Greek words is related to “breathings”, not accent marks. Outside of the digraphs PH, TH and CH, H usually corresponds to a “rough breathing” diacritical mark in Greek.
This diacritical mark was used primarily on vowels at the start of words. It always occurs at the start of a word before the vowel Y; other vowels could take either a “rough breathing” or a “smooth breathing” (indicating the absence of /h/).
In addition, in some texts written in polytonic Greek orthography, the letter rho (corresponding to Latin R) was written with a rough breathing in certain contexts: at the start of a word, and in the middle of the word when it came directly after another rho.
This convention is explained in Nick Nicholas’s answer to the following Quora question: When was it a rule that double rhos (Greek letters - ῤῥ) should be written with smooth and rough breathing marks and when did the rule change?
Nicholas says the following (rearranged so that it fits better with the flow of this answer):
Allen’s Vox Graeca (p. 39) [mentions the evidence and refers to] writing ῤῥ as a Byzantine practice, and it is of course corroborated in the Latin transliteration <rrh> (e.g. Pyrrhus = Πύῤῥος).
The ῤῥ orthography reflects a phonological reality of Classical Greek, that the second rho in a pair was voiceless, something attested in Herodian.
[this practice] had dropped out of use in Modern Greek early in the 20th century. As in fact had the initial rough breathing on rho.
The ῤῥ orthography used to be regular in Western typography, but has long since fallen out of use; from memory, it was routine in early 19th century editions of Classical texts, and rare by late 19th century editions.
In other words, this convention is thought to be based on some phenomenon of using a “aspirated” (in terms of phonetics, possibly just devoiced) sound for rho in certain positions in Classical Greek accents (although in modern Greek there is no difference between aspirated and unaspirated /r/ sounds). And because the conventional way of Latinizing rho with a rough breathing is the digraph “rh”, the conventional way of Latinizing double rho in words from Greek has been “rrh”. See the following blog post on John Wells's phonetic blog: rh and rrh
The word “myrrh”
Apparently the Semitic source of the Greek word μύρρα had a geminate/double/long r (Wiktionary references Arabic murr). The pronunciation of one of the derived Greek words is reconstructed as something like /myrra/ [myr̊r̊a] (in Classical Attic Greek older [u] was fronted to [y]), which was spelled as μύρρα, which was Latinized as myrrha.
The form “murra”, which was spelled without an H, was also used in Latin (in fact, “murra” was apparently the preferred form and spelling of the word in Classical Latin), and the Oxford English Dictionary indicates that “murr-” spellings were used from Old English up into Middle English. So the use of H in the English spelling of this word was inconsistent in the past. There have also been variant spellings with I, as is fairly common for words with the vowel Y from Greek [y].
But the modern standard spelling in English has become fixed as “myrrh”, just as the modern standard spelling in French has become fixed as “myrrhe” (the CNRTL indicates that in the past, alternative spellings like “mirre” existed in French).
- There are many variant Greek alphabets that were used in ancient times.