Using <ui:repeat><h:inputText> on a List<String> doesn't update model values
Solution 1:
The String class is immutable and doesn't have a setter for the value. The getter is basically the Object#toString() method.
You need to get/set the value directly on the List instead. You can do that by the list index which is available by <ui:repeat varStatus>.
<ui:repeat value="#{mrBean.stringList}" varStatus="loop">
<h:inputText value="#{mrBean.stringList[loop.index]}" />
</ui:repeat>
You don't need a setter for the stringList either. EL will get the item by List#get(index) and set the item by List#add(index,item).