I have created an app for contacts insert and delete 
You can fork on github-> xxbinxx/phoneGap-ContactsApp-Android. you can definitely solve your issue after it.
I have used contact ID's for the deletion purpose.
Here's the short code...
var app ={
/********************SOME OTHER CODE*************************/
openContacts: function() {
app.initialize();
var options = new ContactFindOptions();
options.filter = "";
options.multiple = true;
var fields = ["*"]; //"*" will return all contact fields
navigator.contacts.find(fields, app.onSuccess, app.onError, options);
},
// Write contacts in DOM
onSuccess: function(contacts) {
var li = '';
$.each(contacts, function(key, value) {
if (value.name) {
$.each(value.name, function(key, value) {
if (key === 'formatted') {
name = value;
}
});
}
if (value.note) {
note = value.note;
}
if (value.id) {
id = value.id;
}
console.log("id : " + id + "-> name : " + name + " -> note : " + note);
li += '<li style="text-decoration:none;"><b>Name</b>: ' + name + '<div class="removeIcon pullRight" onclick="app.removeThisContact(\'' + id + '\',\'' + name + '\')"> </div><br><b> Note:</b> ' + note + '</li>';
}); // NOTICE the ID is passed as PARAMETER to remove specific contact.
$("#contact").html(li);
},
onError: function(contactError) {
alert('onError!' + contactError.code);
},
removeThisContact: function(id, name) {
console.log("removing contact : " + name);
options = new ContactFindOptions(); // find the contact to delete
options.filter.id = id;
options.multiple = "true";
var fields = ["displayName", "name"]; // you can take any..
navigator.contacts.find(fields, deleteThis, app.onError, options);
function deleteThis(contacts) {
var contact = contacts.pop();
// logging things to troubleshoot.
console.log('inside deleteThisContact: parameter passed: '+ contacts);
console.log("popped out:" +contact);
contact.remove(function() {
$('#status-area')
.flash_message({
text: 'Contact Removed!',
how: 'append'
});
app.openContacts();
}, null);
},
deleteAllTheContacts: function() {
var deleteContact = function(contacts) {
console.log("length = " + contacts.length);
// No contacts left, stop saving
if (contacts.length == 0) {
console.log("All contacts removed");
return;
}
var contact = contacts.pop();
contact.remove(function() {
deleteContact(contacts);
}, null);
};
navigator.contacts.find(["*"], deleteContact, app.onError, {
"multiple": true
});
},
/********************SOME OTHER CODE*************************/
}
$.fn.flash_message = function(options) {
//flash your message
}
Hope this will help you. :)