Éloïse

A diaeresis should be written above the "i" in the French form of this name to signify that it is pronounced as a different letter, otherwise it would sound like [ɛlw¨»:z].
[noted -ed]

The above poster is right--to be pronounced as "ay-lo-EEZ", it needs to be written as "Éloïse". "Éloise", according to the rules of French pronunciation, would be pronounced "el-WAZ".