To get the message as you expected, then you will have to write the code like this
<asp:HiddenField ID="HiddenField1" Value="some stuff" runat="server" />
Since you are assigning the value to the hidden field in Client Side, the value can be accessed only on the
PostBack
if you are assigning the value to hidden field in the
mark up
then you can get the value on Page Load ( not postback )
change the value as below to see how it works
document.getElementById('<%=HiddenField1.ClientID%>').value = "some other more stuff";
Now while the page is loaded for first time you will see the output as
"The input value is :some stuff"
after post back the output will be
"The input value is "some other more stuff"