Click the speaker icon (next to the "all rooms" button) to bring up the volume slider.

SOchat_ping_volume_slider.user.js

// ==UserScript==
// @name         SOchat ping volume slider
// @description  Adds a slider that lets you adjust the volume of the ping notification sound
// @version      1.0
// @author       Paul Pinterits
// @match        *://chat.stackoverflow.com/rooms/*
// @grant        GM.getValue
// @grant        GM.setValue
// @namespace    Aran-Fey
// @require      https://github.com/Aran-Fey/userscript-lib/raw/60f9b285091e93d3879c7e94233192b7ab370821/userscript_lib.js
// ==/UserScript==


(function(){
    'use strict';

    const audioPlayer = document.getElementById('jp_audio_0');
    
    async function loadVolume(){
        const volumeString = await UserScript.getValue('volume', null);
        if (volumeString === null){
            return;
        }
        
        const volume = JSON.parse(volumeString);
        
        audioPlayer.volume = volume;
    }
    
    function saveVolume(volume){
        const volumeString = JSON.stringify(volume);
        
        UserScript.setValue('volume', volumeString);
    }

    function createVolumeSlider(){
        const audioIcon = document.getElementById('sound');
        audioIcon.addEventListener('click', _waitForNotificationDialog);
    }
    function _waitForNotificationDialog(){
        setTimeout(_createVolumeSlider, 100);
    }
    function _createVolumeSlider(){
        const dialog = document.querySelector('div.popup');

        const slider = document.createElement('input');
        slider.type = 'range';
        slider.id = 'ping-volume-slider';
        slider.name = 'ping-volume-slider';
        slider.min = 0;
        slider.max = 1;
        slider.step = 0.01;
        slider.value = audioPlayer.volume;

        slider.addEventListener('change', function(){
            const volume = slider.value;
            
            audioPlayer.volume = volume;
            saveVolume(volume);
        });

        const label = document.createElement('label');
        label.id = 'ping-volume-label';
        label.for = 'ping-volume-slider';
        label.textContent = 'Volume';
        label.appendChild(slider);

        dialog.appendChild(label);
    }

    loadVolume();
    createVolumeSlider();
    add_style(`
#ping-volume-label {
    margin-top: 1em;
    display: flex;
    align-items: center;
    
}

#ping-volume-slider {
    min-width: 0;
    margin-left: 0.5em;
}
`);
})();