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The problem is that India is a large country and speak languages from a number of language families. The pronounciations of the Sanskrit words vary by regions, as are their transliterations into English. The spelling in the native language (even when it is Thai or Laotian: basically all over South Asia) maintains the original Sanskrit spelling to a large extent, but it is often difficult looking at the English spelling unless you know what linguistic family it came through. Mauritius is particularly difficult because both North Indians and South Indians populated it in largish enough numbers: I was hoping you knew what Indian language your father spoke, if any.

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In any case, there is a Sanskrit word daviSTha which is the superlative form of dUra meaning far (i.e. it means farthest, or very far away). The exact origin of the word is not clear: there is an obscure root du/dU, with cognates in Greek, which technically means to go (not to be confused with a homophone meaning to burn), and these words may be related. The d- in this word would be like th in English the, the v somewhere between English v and w, the S is retroflex and the Th is retroflex aspirated. This is what in South Indian English transliteration would become dhavishta, which is what you write. The word is very old: it appears already in the Rig veda.

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If the name came from North Indian transliteration, then I do not quite know: nothing fits exactly, so I suspect it is indeed South Indian.