[SalesForce] Dynamically changing lightning button icon name with setAttribute not working

In this line of code, this path should refer to your SLDS library which you uploaded in Static resources.

<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="/assets/icons/utility-sprite/svg/symbols.svg#search"></use>

For example:

<use xlink:href="{!URLFOR($Resource.SLDS100, '/assets/icons/custom-sprite/svg/symbols.svg#custom3')}" />

<aura:iteration items={!v.mydata} var ="obj" indexvar="index">
<tr id={!index}>
<td>    <lightning:buttonIcon iconName="{!obj.icon}" title="Default" 
           alternativeText="Default"
                      onclick="{!c.add}" /></td>
<td class="" data-label="Name">{!obj.Name}</td>
<td class="" data-label="Type">{!obj.Type}</td>
<td class="" data-label="Age">{!obj.Age}</td>
</tr>
</aura:iteration>

You have stored icon in single attribute (v.icon) which changes for each and every iteration of your object.

So you can store icon name in your v.mydata variable

init : function(component){
    var data = component.get('v.data');
    data.map(function(item){
         item.icon = 'utility:add';
         return item;
    });
}

Now onclick

add: function(component, event, helper) {
   var index = event.target.id;
   var data = component.get('v.mydata');
   var item = data[index];
   item.icon = "utility:add" ? "utility:check" : "utility:add";

   component.set("v.mydata",data); 

}