Because of the awful disaster in Haiti, that country's name has been in the news, and this prompted reader RH to wonder about its English pronunciation.
I am currently watching the CNN coverage of the earthquake in Haiti, and without exception the name of the country is pronounced "hatey". Surely "Hayiti", or even "Ayiti" would be better, and just as easy for the reporters to learn?
In IPA, the observation would be that American English speakers generally say [ˈɦeɪ.ɾi], whereas a closer imitation of French (or Haitian) pronunciation would be something like [ɐ.iˈti].
But the conventional English version of foreign names is often — even usually — quite far away from the native's version. Italy is not very close to the way that Italians pronounce Italia; Russia is similarly distant, in stress pattern and vowel quality, from Россия; and in cases like Germany for Deutschland or Finland for Suomi, there isn't even a morphological connection.
In fact, there are relatively few countries whose conventional English name is the closest possible English approximation to the native pronunciation. (And this is complicated by the existence of many countries with more than one official language, and thus more than one official name, like die Schweiz / la Suisse / la Svizzera / la Svizra, none of which are optimally approximated phonologically by "Switzerland".)
I haven't been able to determine when the current English pronunciation of Haiti became conventionalized, but I imagine that it goes back as far as the name does. I guess that this would be the early 19th century — the name was adopted in 1804 at the time Haitian independence from France, from "one of the indigenous Taino names for the island".
[While you're thinking about it, why not donate generously to Haitian disaster relief? Some information for Americans about how to do this can be found here. For example, if you text "HAITI" to 90999, a donation of $10 will go to the Red Cross for relief efforts, charged to your cell phone bill. I'm sure that the rest of you can easily find your local alternatives.]