You should keep the select element disabled but also add another hidden input with the same name and value.

If you reenable your SELECT, you should copy its value to the hidden input in an onchange event and disable (or remove) the hidden input.

Here is a demo:

$('#mainform').submit(function() {
    $('#formdata_container').show();
    $('#formdata').html($(this).serialize());
    return false;
});

$('#enableselect').click(function() {
    $('#mainform input[name=animal]')
        .attr("disabled", true);
    
    $('#animal-select')
        .attr('disabled', false)
    	.attr('name', 'animal');
    
    $('#enableselect').hide();
    return false;
});

#formdata_container {
    padding: 10px;
}

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<div>
    <form id="mainform">
        <select id="animal-select" disabled="true">
            <option value="cat" selected>Cat</option>
            <option value="dog">Dog</option>
            <option value="hamster">Hamster</option>
        </select>
        <input type="hidden" name="animal" value="cat"/>
        <button id="enableselect">Enable</button>
        
        <select name="color">
            <option value="blue" selected>Blue</option>
            <option value="green">Green</option>
            <option value="red">Red</option>
        </select>

        <input type="submit"/>
    </form>
</div>

<div id="formdata_container" style="display:none">
    <div>Submitted data:</div>
    <div id="formdata">
    </div>
</div>

^{Note: This is an updated answer. Comments below refer to an old version which messed around with keycodes.}

JavaScript

Try it yourself on JSFiddle.

You can filter the input values of a text <input> with the following setInputFilter function (supports Copy+Paste, Drag+Drop, keyboard shortcuts, context menu operations, non-typeable keys, the caret position, different keyboard layouts, and all browsers since IE 9):

// Restricts input for the given textbox to the given inputFilter function.
function setInputFilter(textbox, inputFilter) {
  ["input", "keydown", "keyup", "mousedown", "mouseup", "select", "contextmenu", "drop"].forEach(function(event) {
    textbox.addEventListener(event, function() {
      if (inputFilter(this.value)) {
        this.oldValue = this.value;
        this.oldSelectionStart = this.selectionStart;
        this.oldSelectionEnd = this.selectionEnd;
      } else if (this.hasOwnProperty("oldValue")) {
        this.value = this.oldValue;
        this.setSelectionRange(this.oldSelectionStart, this.oldSelectionEnd);
      } else {
        this.value = "";
      }
    });
  });
}

You can now use the setInputFilter function to install an input filter:

setInputFilter(document.getElementById("myTextBox"), function(value) {
  return /^\d*\.?\d*$/.test(value); // Allow digits and '.' only, using a RegExp
});

See the JSFiddle demo for more input filter examples. Also note that you still must do server side validation!

TypeScript

Here is a TypeScript version of this.

function setInputFilter(textbox: Element, inputFilter: (value: string) => boolean): void {
    ["input", "keydown", "keyup", "mousedown", "mouseup", "select", "contextmenu", "drop"].forEach(function(event) {
        textbox.addEventListener(event, function(this: (HTMLInputElement | HTMLTextAreaElement) & {oldValue: string; oldSelectionStart: number | null, oldSelectionEnd: number | null}) {
            if (inputFilter(this.value)) {
                this.oldValue = this.value;
                this.oldSelectionStart = this.selectionStart;
                this.oldSelectionEnd = this.selectionEnd;
            } else if (Object.prototype.hasOwnProperty.call(this, 'oldValue')) {
                this.value = this.oldValue;
                if (this.oldSelectionStart !== null &&
                    this.oldSelectionEnd !== null) {
                    this.setSelectionRange(this.oldSelectionStart, this.oldSelectionEnd);
                }
            } else {
                this.value = "";
            }
        });
    });
}

jQuery

There is also a jQuery version of this. See this answer.

HTML 5

HTML 5 has a native solution with <input type="number"> (see the specification), but note that browser support varies:

• Most browsers will only validate the input when submitting the form, and not when typing.
• Most mobile browsers don't support the step, min and max attributes.
• Chrome (version 71.0.3578.98) still allows the user to enter the characters e and E into the field. Also see this question.
• Firefox (version 64.0) and Edge (EdgeHTML version 17.17134) still allow the user to enter any text into the field.

Try it yourself on w3schools.com.