My tefillin shel yad had a cover that could slip on and off but the cover is lost and needs replacement. Is there a way I can determine the size so I can ask for a replacement? Are there standard sizes?
Yes, there are standard sizes, though obviously there are usually going to slight variations in fit for every pair of handmade tefilin. The sizes seem to be measured by
cm2 millimeters. Based on my experience they tend to err towards a looser fit; I usually adjust this by adding a sticker or a piece of paper on the inside.
This piece is sometimes called a "yadlich," and a quick Google brings me to http://keterjudaica.com/Yadlich-Tefillin-Holder.html
(There's another page that uses the same size values but claim them to measure "cm.")