TYPO3 10.4: How can you add another dropdown field to the appearance tab of an content element?

You can't do this via the backend. You will need to extend the TCA (e.g. in your sitepackage extension).

In your extension:

1. Add the field in ext_tables.sql
2. Extend the TCA in Configuration/TCA/Overrides/tt_content.php:

• add new fields to the columns
• Add the new fields to the visible fields
• Add the new fields to the appropriate palette

Tip: You can look at the active TCA configuration for tt_content in the TYPO3 backend Configuration | Globals['TCA']

If you want to place the new field after an existing field, e.g. space_before_class, check which palette this is in ('frames') and place it in that.

Example:

ext_tables.sql:

CREATE TABLE tt_content(
  my_new_field TINYINT(1) UNSIGNED DEFAULT '0' NOT NULL
);

Configuration/TCA/Overrides/tt_content.php

$fields = [
    // new field
    'my_new_field' => [
        // title of the field 
        // alternatively, just use a string, but putting the string in
        // language file eases translations 
        'label' => 'LLL:EXT:myextension/Resources/Private/Language/locallang.xlf:tt_content.my_new_field.title',
        'config' => [
            // type: select 
            'type' => 'select',
            'renderType' => 'selectSingle',  
            'default' => 0,
            'items' => [
                [
                    // again, better to use language label LLL here 
                    'description',                         
                    // value
                    0
                ],
                [
                    // another option
                    'description 2',
                    // value
                    1
                ],
            ],
        ],
    ],
    // ...
];


// Add new fields to table:
\TYPO3\CMS\Core\Utility\ExtensionManagementUtility::addTCAcolumns('tt_content', $fields);
     
// Add new field to palette 'frames' 
\TYPO3\CMS\Core\Utility\ExtensionManagementUtility::addFieldsToPalette(
    'tt_content',
    'frames',
    'my_new_field',
    'before:frame_class'
);

Resources:

• extend TCA: https://docs.typo3.org/m/typo3/reference-coreapi/main/en-us/ExtensionArchitecture/HowTo/ExtendingTca/Examples/Index.html
• TCA: columns field types: https://docs.typo3.org/m/typo3/reference-tca/main/en-us/ColumnsConfig/Index.html
• select types: https://docs.typo3.org/m/typo3/reference-tca/main/en-us/ColumnsConfig/Type/Select/Index.html
• For the rest of the options, refer to the TCA Reference: https://docs.typo3.org/m/typo3/reference-tca/main/en-us/Index.html

If you could live just with more options to the given fields you can do it in the BE without programming / extending the TCA (Sybille's answer).

Extending the given list of select options for the existing fields is possible with some page/user TSConfig like:

TCEFORM.tt_content.layout.addItems {
    20 = additional layout 1|nyt layout 1|neues Layout 1|...
    25 = additional layout 2|nyt layout 2|neues Layout 2|...
}
TCEFORM.tt_content.frame_class.addItems {
    50 = frame-class-50
    55 = frame-class-55
}

Of cause this needs a handling in the rendering of the CEs.
Depending on the TYPO3 version and the active rendering there are different places to handle this.
Without FLUID rendering it needs some edits in the typoscript, with FLUID you need to override the appropriate file (layout/template/partial) where the field (here: layout and frame_class) is handled.