You need to clarify as to in which format does SearchSynonym return the IDs and NAMEs? Is it in JSON? If it is in JSON, then try the following:
var companyList = $("#Company_List");
$("#company").change( function(){
$.getJSON("/Home/SearchSynonym/",{ query: $(this).val() }, function(response){
var responseList = "";
$.each(result, function(index, item){
responseList += "<option value='" + item.id + "'>" + item.name + "</option>";
});
companyList.html(responseList);
});
});
This would work if your 'company' text field bears the id="company" in the tag decleration, your 'Company_List' dropdown bears the id="Company_List" in the tag decleration and your server end receives the parameter "query" for pulling records.