The 'hidden' hinge on most cabinets is often referred to as a "Blum hinge" (pronounced 'bloom'), because that manufacturer made popular a machine for boring those cup holes for the hinges, and manufactures the hinges themselves.
These hinges can usually be adjusted a small amount, to help square up doors, etc.
The 'drill bit' most often used for these types of hinges is a 35, or 40 mm forstner bit, in either the boring machine made for these hinges, or in a drill press, though you can get a jig to do it by hand. (Don't forget, you don't want to drill all the way through, heheh)
The problem you will have is the void left by the original hinge placement... you may actually want to consider switching to a different hinge altogether.
Here are some links to check out -
http://www.rockler.com/product.cfm?page=1633&filter=hinge%20bit
http://www.rockler.com/product.cfm?page=2406
http://www.rockler.com/CategoryView.cfm?Cat_ID=45
I would seriously recommend that you talk to a cabinet installer, or small cabinet shop. Either of those would already own the necessary tools, and if necessary, have the ability to make a new piece of trim, or even a new door, to match your existing cabinetry, if relocating the hinge is a problem.
Have Fun