You will need an opening select tag that tells PHP it is an array. Something along the lines of the following should get you on your way…
<?php
// Display Fields
add_action( 'show_user_profile', 'add_multiple_choice_dropdown ' );
add_action( 'edit_user_profile', 'add_multiple_choice_dropdown ' );
function add_multiple_choice_dropdown ( $user ) {
$current_selections = get_user_meta( $user->ID, 'multi_dropdown', true );
?>
<h3>Extra profile information</h3>
<table class="form-table">
<tr>
<th><label for="multi_dropdown" multiple="multiple">The dropdown with multiple choices</label></th>
<td>
<select name="multi_dropdown[]">
<option value="first_choice" <?php echo ( !empty( $current_selections ) && in_array( 'first_choice', $current_selections ) ? ' selected="selected"' : '' ) ?>>First Choice</option>
<option value="second_choice" <?php echo ( !empty( $current_selections ) && in_array( 'second_choice', $current_selections ) ? ' selected="selected"' : '' ) ?>>Second Choice</option>
<option value="third_choice" <?php echo ( !empty( $current_selections ) && in_array( 'second_choice', $current_selections ) ? ' selected="selected"' : '' ) ?>>Third Choice</option>
</select>
<p class="description">Choose from the options above.</p>
</td>
</tr>
</table>
<?php
}
// Save fields
add_action( 'personal_options_update', 'save_multiple_choices' );
add_action( 'edit_user_profile_update', 'save_multiple_choices' );
function save_multiple_choices( $user_id ) {
if ( isset( $_POST['multi_dropdown'] ) ) {
update_user_meta( $user_id, 'multi_dropdown', $_POST['multi_dropdown'] );
}
}
?>