I want to save as jpg with the same name as the selected layer.

I need to turn on the visibility of that layer, save the file with the name of that selected layer.

Then turn manually some layers off and some other layers on. Then select again a layer to get the file name for the save and so on.

By @Dusan331314882hci

Try this script:

/*
Save JPEG Using Active Layer Name.jsx
v1.1, 29th March 2024, Stephen Marsh
Based on:
https://community.adobe.com/t5/photoshop-ecosystem-discussions/export-jpg-and-combining-the-visual-layer-names-to-create-the-file-name/td-p/13617072
*/

#target photoshop

(function () {
    // Set the doc and path variables
    var doc = app.activeDocument;
    var docName = doc.name.replace(/\.[^\.]+$/, '');
    try {
        var docPath = doc.path.fsName;
    } catch (e) {
        var docPath = Folder.selectDialog("Unsaved base file, select the output folder:");
    }

    // Clean the layer name, place characters to be retained inside the [] brackets...
    var layerName = doc.activeLayer.name.replace(/[^a-z0-9 -_]/gi, '').replace(/^ +| +$/g, '');
    var jpgSave = new File(docPath + "/" + docName + "" + layerName + ".jpg");

    // Check for existing file
    if (jpgSave.exists) {
        // true = 'No' as default active button
        if (!confirm("File exists, overwrite: Yes or No?", true))
            return;
    }

    // Setup the save options
    var jpgOptns = new JPEGSaveOptions();
    jpgOptns.formatOptions = FormatOptions.STANDARDBASELINE;
    jpgOptns.embedColorProfile = true;
    jpgOptns.matte = MatteType.NONE;
    jpgOptns.quality = 12;
    doc.saveAs(jpgSave, jpgOptns, true, Extension.LOWERCASE);
}());

Note:

Layers can have special characters that are illegal when used as a filename.

In the code you will find the following section, simply add any special characters that you wish to retain inside the [ ] square brackets:

[^a-z0-9 -_]

The code currently retains alpha-numeric characters, digits, word spaces, hyphens and underscore characters.

If you wished to add a $ sign, then you would do so similar to this:

[^a-z0-9 -_$]

Keep in mind that Photoshop may still remove illegal characters anyway.