ID's must be unique, but you could save some time and add a parameter to your function that takes the current element being clicked:
onclick="changestream1(this);"
function changestream1(obj)
{
var streamInput = obj.value;
var object = document.getElementById('live_embed_player_flash');
var param = document.getElementById('param');
object.setAttribute("data", "http://www.twitch.tv/widgets/live_embed_player.swf?channel=" + streamInput);
param.setAttribute("value", "hostname=www.twitch.tv&channel=" + streamInput + "&auto_play=true&start_volume=25");
}