You'll have to remove the checkbox for Allow New Values for dialog fields so the choices show up on the web.
If you don't want to use JavaScript, you can add an "Other" field (plain text input field), then in the input translation formula for the main dialog field, append this Other field if it is not empty.
If you can use JavaScript (thanks to Michael Holmstrom):
- add a "var frm;" to the forms JS Header-event
- add a "frm=window.document.forms[0];" to the forms onLoad-event
- add a button "New value" with the following JS-code:
var response= prompt("Please add your new value",'');
if (response !=null) {