Set an onclick handler for the button and then have it trigger a javascript function - note the use of the alert - just to show the functioning of the function:

function promptMe(){
    var userAdjective = prompt("Please provide an Adjective");
    alert (userAdjective);
}

<button id="bgnBtn" onclick="promptMe()">Start Game</button>

You could use jQuery and do a 3 button dialog(). Check out this working jsFiddle demo:

$("#dialog").dialog({

    autoOpen: true,
    buttons: {

        Yes: function() {

            alert("Yes!");
            $(this).dialog("close");
        },
        No: function() {

            alert("No!");
            $(this).dialog("close");

        },
        Maybe: function() {

            alert("Maybe!");
            $(this).dialog("close");
        }

    },
    width: "400px"

});