Why is the spelling "eureka" by far more preferable to "heureka" in English? Greek vocabularies give "heureka" for the perfect to "heurisko".
As the Oxford English Dictionary says, 'the correct spelling heureka is rare'. Heureka is found in the dictionary’s citations only until 1806. Twelve years later, even that great hellenophile Byron writes in 'Childe Harold':
I don’t know what makes the spelling ‘Heureka’ the correct one. The Greek is εὕρηκα, which has the same initial vowel as Εὐρῑπίδειος (Euripides). Well, not quite the same because it has a different diacritic mark and therein may lie some sort of explanation, perhaps to do with Greek aspiration (in the phonetic sense). We need a Greek scholar to tell us more.
Eureka is the exact transcription of the Greek word in the Latin alphabet. The additional h in the correct written form marks the pronunciation, which in ancient Greek was denoted with punctuation signs right above the letter u. This pronunciation has now become extinct in Greek, although this word is still used.
My guess is that when eureka started being widely used in the English speaking world, the pronunciation of the word in Greek had already changed, so there was nothing to be denoted with an additional h in the beginning of the word. Therefore, even erudite speakers adopted the simpler form of writing.
In summary, my hypothesis is that certain translators transliterated the Greek letters but not the accents, and thereby lost the [h], because initial [h] is only a small mark above the vowel. They may have done so because their Greek was not very good, or because they did not consider the [h] mark a real letter worth transliterating.
The Greek word εὕρηκα was pronounced [hěu̯rɛːka] in classical Greek; the [h] is indicated by the spiritus asper ("rough breathing"), the tiny "c" above the ὑ. (The other accent, the acutus, indicates increasing pitch; note that the spiritus asper came from a tiny left half of the Greek letter H, which was used in some older dialects to denote an [h] sound.)
Many, perhaps most Greek words came to English through Latin. However, the Latin transliteration of εὕρηκα would be heureca, or possibly eureca. My own, quick research in the Bibliotheca Teubneriana Latina suggests that the great majority of Roman writers transcribed the spiritus asper as h, but a significant minority did not. In any case, neither heureca nor eureca exists in the BTL; therefore I suspect that the word was not the famous expression in Rome that it is now.
The earliest mention of Archimedes' exclamation is by Vitruvius (De Architectura IX, preface 10), and later usage is most probably all based on him. Both the Perseus Digital Library, whose edition of Vitrivius is based on the 1912 Teubner, and the edition by C. Fensterbusch (Darmstadt 1964/76) give the word in Greek letters. Perseus/Teubner give the Greek with accents, Fensterbusch without.
The first use of the word recorded by the Oxford English Dictionary is from 1570:
Notice that the word is given in Greek capitals, without accents.
(The square brackets may indicate either something added by editors in a quotation (unlikely, because that only applies to parts of quoted sentences), or a "spurious" entry in the dictionary, which is a word that erroneously found its way into other dictionaries while in fact it was never used (just as unlikely). I conclude that the brackets either indicate nothing of relevance or that this is a dubious quotation.)
By the 19th century, eureka seems to have prevailed. Before that, as it can be seen, various versions were used. Several editions did not print the breathing mark: perhaps this caused readers who were not well versed in Greek to translate the Greek letters one by one into the most similar English letters, disregarding conventions; this would give eureka. A commenter on Languagehat gives the following for the editions of Fielding also quoted in the OED:
This might suggest that the spelling eureka did indeed originate in a letter-by-letter transliteration of the Greek into similar English letters by someone who didn't know the traditional convention of using c for Greek κ: Fielding feared that his version Eureka might seem illiterate and changed it into Heureka in the third edition.
In the time of Vitruvius (1st century BC), the spiritus was most probably not yet in use, so that plain unaccented Greek letters would have been written. It would have been capitals, because Greek minuscules were probably not yet used either. (Note that we cannot be sure that the Greek wasn't added later by monks and that Vitruvius himself didn't use Latin letters.) In the edition by Fensterbusch mentioned above, the Greek letters printed without spiritus too. This might support my theory. But we would really need to compare many more editions from various dates.
So nothing is certain. Before the 19th century, transliterations of Greek words that were not used in Latin varied wildly—if not of other Greek words, then certainly of this one. The reason why it eventually became eureka in modern English, Dutch, and French, as opposed to heureka in German, could be any arbitrary circumstance, though I have a feeling that the German version is relatively new as compared to ours. In the edition of De Architectura below, printed in 1567 in Venice, the word is written in Latin letters as eurica:
This shows how varied transliterations were in the 16th century. The i can probably be explained by the fact that it was printed in Italy. But here is the Latin edition of Giambattista della Porta's Magia Naturalis from 1597 (the English version is in the OED's quotation from 1658 above):
It was printed in Frankfurt, and eureka is given in Greek, with spiritus. Another Latin edition of the same work, printed in Naples in 1589, uses the aspirated Greek too (scanned image no. 302 / printed page 285, at the bottom).