These days we usually use picks made from plastic.  They last longer and are easy to find.  There are still picks made of horn for example, and some still use feather picks as were used in the past, but with all the varieties of plastic available, at various thicknesses and stiffness, for me at least there is no reason to look elsewhere. 
I've been told when using horn picks, they should be soaked in light oil (not olive oil) to make them more pliable and longer lasting.
Even when you buy a pick (all the oud makers I've listed make/sell picks) it usually will need some touching up by you to customize it to your taste.  Use very fine sandpaper to do this, make sure you leave no sharp edges, make it as smooth as possible.  You want the least amount of pick noise as you play.  To do the last touchups you can use a fine nail file or even just rub the end on your carpet, just make sure that whenever you are sanding/rubbing down your pick, you stop when you feel the material getting warm.  The plastic can become chronically 'floppy' if you keep working on it as it gets hotter.
If you want to make your own, some places to look for good plastic material are:  A plastics shop of course, the white handles on European style shopping bags that clip together, comb spines, and my personal favorite, heavy duty cable ties from an electrical supply store. 

 Some examples:

You should realize that your pick is a thing of very personal preference, make it comfortable and what you need to make the kind of sound you love.  A pick that is perfect for me in all likelihood won't be for you.

Copyright Mavrothi T. Kontanis. All rights reserved 2008