Not sure where you’re adding this functionality, but I don’t see anywhere in this code where you use the update_option() function. Above your form HTML you should check if the form has posted data, then update the option.
//Do nonce checking here and
$tweak = $_POST['tweakfunction1'] ? $_POST['tweakfunction1'] : '';
update_option('tweakfunction1', esc_html($tweak));
WordPress includes a function called checked() that checks against a certain value. Your input tag has no value attribute as well. I added it below with the checked function.
<input size="76" name="tweakfunction1" type="checkbox" id="tweakfunction1" <?php checked('foobar', get_option('tweakfunction1'));?> value="foobar" />