Hm... That's a hard one. I tryed myself and it did arrange them differently, but not by name - lol. I think if you rename things like recycle bin, it doesn't use that name, but uses the original recycle bin name. I renamed mine Filthy Garbage and it always comes up last or beginning when I arrange by name. You should be able to manually arrange your icons, and right click and make sure auto arrange is NOT selected. Then, after you arrange your icons, right click - refresh. It should stay that way from that point on, but I have done this before and every so often it screws up the arrangement a little bit.
Other than that, if you try renaming all your icons in alpha.. order, then clicking on arrange by date modified, that might work as well. That's all I can think of for now! Sorry!
Good luck though.