"Delphi" originates from Greek
The Greek name is written Δελφοί. This is a plural form, interestingly enough.
The reconstructed Classical pronunciation of this is, as mechanical snail mentions, [delpʰo͜í]. The nearest thing to that in English would be /dɛlˈpoɪ/ or /deɪlˈpoɪ/. Nobody says this.
The Modern Greek pronunciation is, as "note to self"'s answer mentions, [ðe̞lˈfi]; the nearest thing to that in English would be /ðɛlˈfi/. Nobody says this. Some people do say /ˈdɛlfi/, which is kind of similar but has a different first consonant and also a different stress pattern.
In English, the "i" in "Delphi" is pronounced as the Latin plural suffix "-i"
In fact, it’s easier to predict the pronunciation of the word if you just ignore the Greek spelling and pronunciation entirely and look at it as a Latin word. The way “Delphi” is spelled indicates that the word came into English through Latin (like many English words that are ultimately of Greek origin): if it had been taken directly into English as a transliteration of the (ancient) Greek spelling, we’d expect it to be spelled “Delphoi”, and if it had been taken directly into English as a transcription of modern Greek pronunciation, we’d expect it to be spelled “Dhelfi” or at least “Delfi”.
In “Delphi” the Greek plural suffix “-οι” has been replaced with the (cognate) Latin plural suffix “-i". In English words, the Latin-derived plural suffix -i is pronounced as either /aɪ/ or /i:/; neither is generally considered incorrect. Neither pronunciation is restricted to one part of the English-speaking world either: the Oxford English Dictionary entry for “Delphi” (in the sense “Delphi forecasting”) gives /ˈdɛlˌfaɪ/ as the first pronunciation for both British and American speakers.
Bilingual speakers may be more likely to use /i/
The pronunciation of the Latinate suffix "-i" as /aɪ/ is the result of the English-specific Great Vowel Shift; in other European languages, this kind of shift of original long /i/ to /aɪ/ either did not take place (e.g. in the Romance languages, like French or Italian), or was inhibited or reverted in Latinate words because of spelling (I'm thinking here mainly of German and Dutch dialects, many of which did undergo a similar vowel shift but spell the resulting diphthong not with the letter "i", but with digraphs like "ei" or "ij"). This means that non-native speakers of English who speak one of these other European languages (I assume this is the case for Marco Cantù, the programmer mentioned by Peter Turner) may tend to use /i:/ because it is more similar to the pronunciation used for the letter "i" in their native language.