I want to change tha arrow of a select dropdown box (the one in blue on the right)

Try this :
option:hover{
background-color:#00FF00;
}

Result may be odd with this color code, but I think this is what you are looking for. It will certainly work for Fx, but I don't know for IE.

Mr.agthorpe in this way u can change the cursor and also define the hover so that it can automatically change when cursor move on it.
div.cursor{
cursor:crosshair;
}