If you look at the screen capture above, you will see that the Web Part Title exists in a SPAN included in an H2 with the class ms-webpart-titleText. If you are using jQuery, you could do something like the code below to find the specific web part.
var webpart;
var webPartTitle = "Some Specific Web Part";
$("h2.ms-webpart-titleText").each(function()
{
$(this).find("span:contains('"+webPartTitle+"')").each(function()
{
if ($(this).text() == webPartTitle){
webPart = $(this).closest("span").closest("[id^='MSOZoneCell_WebPart']");
}
}
}
Most of the time, there will only be one SPAN that contains the title but use the .each in case SharePoint decides to do something different. That should get you the specific web part and then you can change whatever you want.