DataWrapper.cls

public with sharing class DataWrapper {
    public Object value { get; set; }

    public DataWrapper() {}
    public DataWrapper(Object theValue) {
        value = theValue;
    }
}

NameInputTextComponent.component

<apex:component controller="NameInputTextComponentController">
  <apex:attribute name="value" assignTo="{!theValue}" description="Wrapped value" type="DataWrapper" required="true" />
    
  <apex:form >
    <apex:inputText value="{!name}" id="__inputTextComponent">
      <apex:actionSupport event="onblur" action="{!formatName}" reRender="__inputTextComponent" />
    </apex:inputText>
  </apex:form>
</apex:component>

NameInputTextComponentController.cls

public with sharing class NameInputTextComponentController {
    public NameInputTextComponentController() {}

    public DataWrapper theValue { get; set; }

    public String name {
        get {
            return (String)theValue.value;
        }
        set {
            System.debug('the new value : ' + value);
            theValue.value = value;
        }
    }

    public PageReference formatName() {
        List<String> splits = name.split(' ');
        List<String> capitalizedSplits = new List<String>();
        for (String s : splits) {
            capitalizedSplits.add(s.capitalize());
        }
        name = String.join(capitalizedSplits, ' ');
        return null;
    }
}

NameInputTextPage.page

<apex:page controller="NameInputTextPageController" showHeader="true" sidebar="true">
  <c:NameInputTextComponent value="{!theName}" id="nameComponent" />

  <apex:pageMessages id="messages" />

  <apex:form>
    <apex:commandButton action="{!empty}" reRender="nameComponent, messages" value="Rerender" />
  </apex:form>
</apex:page>

NameInputTextPageController.cls

public with sharing class NameInputTextPageController {
    public NameInputTextPageController() {}

    public DataWrapper theName {
        get {
            if (theName == null) {
                theName = new DataWrapper('First Name');
            }
            return theName;
        }
        set;
    }

    public PageReference empty() {
        ApexPages.addMessage(new ApexPages.Message(ApexPages.Severity.INFO, (String)theName.value));
        return null;
    }
}