Many Spanish words taken into English have a "J" sounding like "H", but San Jacinto follows a different rule:
Why is San Jacinto not pronounced San Hacinto in English?
The languagehat article mentioned in comments shows that the J in Jacinto is under attack in Texas, but it doesn't explain why Texans don't pronounce Jacinto like every other Spanish J word.
Rules of pronunciation tend to mean less when Proper Nouns come into play. Largely, it's a matter of tradition. "Jacinto", as a Spanish word, means "hyacinth". If you were speaking about the flower, you would pronounce it [ha-SEEN-to]. However, in the context of "San Jacinto", the pronunciation becomes [juh-SIN-to].
Why? Because that was how the original settlers of that city pronounced it. The name became canonized, and remains in that (technically incorrect) pronunciation as a matter of tradition. That pronunciation became associated with that location, changing it could lead to confusion.
When it comes to names, you simply can't count on consistency of pronunciation. Why should Des Moines, IA, be pronounced [deh moyn], but Des Plaines, IL, is instead [des playnz]? Exact same situation.
The earliest Merriam-Webster dictionary to include an entry for (and a pronunciation of) "San Jacinto" is An American Dictionary of the English Language (1864), which provides this brief entry in its "A Pronouncing Gazetteer" appendix to the main dictionary:
San Jacinto (Sän Ja̯-sĭn´-tō) agr[icultural] & past[oral] co[unty] S[outh] E[ast] Tex[as] pop[ulation] 6 [that is, between 5,500 and 6,499], [county seat] Cold Spring.
The gazetteer pronunciation notes report that the pronunciation of the a̯ sound in Jacinto is "obscure." They also clarify that indicates that the ä sound in San matches the a in far.
The entry in the "Pronouncing Gazetteer" in Webster's International Dictionary (1890) is identical to the one in its 1864 predecessor except that the population of San Jacinto County, Texas, is given as 7 (that is, as falling into the range of from 6,500 to 7,499).
The "Pronouncing Gazetteer" in Webster's New International Dictionary (1909), however, has a longer entry for the place name and a slightly modified pronunciation for it:
San Jacinto (Săn Jȧ-sĭn´-tō) m[oun]t[ain] San Bernardino range, S[outhern] Calif[ornia] 10,805 ft. high. — co[unty] E[ast] Texas, 602 [square miles] pop[ulation] 10 [that is, between 9,500 and 10,499], [county seat] Coldspring. — river, Texas, Walker co[unty] to Galveston bay ; battle near mouth 1836.
The a in San has switched from ä as in arm to ă as in am, and the a in the first syllable of Jacinto has switched from "obscure" to ȧ as in ask. It is unclear to me how the ȧ in ask differs from the ä in am, since (I think) I tend to pronounce those vowel sounds identically; but doubtless there is some subtle difference at work that eludes my detection.
The pronunciation information provided in the three major editions of Webster's full-length dictionary from 1864 through 1909 yield several interesting details. The first (and most pertinent to the posted question above) is that the first letter in Jacinto was pronounced as j rather than as h throughout this period, meaning that the standard anglophone Texan pronunciation of the word has favored the j sound for at least 158 years.
Also interesting is the shift in the pronunciation of San, which in the 1864 and 1890 dictionaries roughly replicated the Spanish pronunciation, with the a sounded as in far, but which in 1909 was reported to have the same a sound as am.
A further noteworthy point is that the 1909 Webster's states that the mountain in Southern California is pronounced with the same j sound as the Texas county and river—although site participant Jeshizaemon, in a comment beneath the posted question, reports that Southern Californians now pronounce the first letter in the mountain's name as y rather than as j. It is possible, of course, that the people at Merriam-Webster simply added the California mountain to the gazetteer entry for the Texas county without inquiring into whether the standard anglophone pronunciation of the name was the same in both places.
In contrast, the j pronunciation of Jacinto remains dominant in anglophone Texas and is unlikely to change anytime soon, especially as, from an early age, generations of monolingual anglophone Texas schoolchildren learn about the Battle of San Jacinto (with a j pronunciation) as the decisive engagement in the war for Texas independence from Mexico. I remember hearing an account of the Battle of San Jacinto from my teacher in first or second grade (in Corpus Christi, Texas) and finding it utterly baffling that the general in charge of the Mexican army (Santa Anna) had the same first name as Santa Claus.
I can't say why Texans originally chose to pronounce Jacinto with a j sound rather than an h sound. One possibility is that early literate anglophone Texans read about the 1836 Battle of San Jacinto in letters, newspapers, or books and simply pronounced the word the way it looked to them on paper. Whatever the reason, the preference arose a long time ago and remains the standard pronunciation among anglophone Texans to this day.
Side note: 'javelina' with an 'h'
On a phonetically related note, I observe that the collared peccary, a mammal that looks like a wild pig and is native to south Texas (as well as to other parts of the southwestern United States and to Mexico) is widely known in Texas by its Spanish name javelina—and the standard anglophone pronunciation of the j is as an h sound. Merriam-Webster took notice of javelina much later than it did San Jacinto: javelina debuted Webster's Eighth Collegiate Dictionary (1973)—with the only pronunciation listed as häv-ə-**'**lē-nə. That pronunciation matches the one I have been familiar with from an early age, but I was surprised that MW didn't have an entry for the word much earlier. In Southeast Texas, where I grew up, javelina was the standard term for the animal in everyday speech; peccary was a word used only in books and (more rarely) newspapers. But it appears that javelina managed to travel under the literary and lexicographical radar for a long time.
A search of The Portal to Texas History (a database of Texas newspapers) yields a total of 101 matches for peccary (including a substantial number of duplicates) in Texas newspapers from as early as 1841 to as late as 2010, with a decade maximum of 17 in the period from 1940 to 1949 and with 34 of the total appearing before 1910. A companion search for javelina in the same newspaper database yields 621 matches (again with a substantial number of duplicates) from as early as 1910 to as late as 2012, with 4 matches in the 1910s (versus 10 for peccary), 4 more in the 1920s (versus 3 for peccary), 34 in the 1930s (versus 13 for peccary), 48 in the 1940s (versus 17, as noted above), and 61 in the 1950s (versus 9), building up to a decade maximum of 147 in the 1980s (when the number of matches for peccary was zero). I have not found any explanation for the triumph of javelina over peccary in Texas English despite the latter's significant head start.
In any case, for purposes of this answer, it seems relevant that anglophone Texans were perfectly comfortable, from at least the middle decades of the twentieth century forward, pronouncing javelina with an h at the same time that they pronounced San Jacinto with a j.
