As in any engineering dilemma, the answer to what is “best” is directly related to the specific situation at hand . . . aka, there is no universal best material, only some choices that are better than others for a specific application.
For project boxes, premanufactured enclosures are available in an almost unlimited array of materials, sizes and designs; and usually offer the most professional looking results. Most projects seem to fit in enclosures costing $5-20 and in my experience, despite a well-equipped shop and good access to free or almost free acrylic, polycarbonate and ABS sheet stock I still often buy an enclosure because the materials, hardware and some kind of new milling cutter I always seem to need often outweigh the cost of the enclosure . . . plus my time, plus the improved cosmetics.
Although more expensive, polycarbonate is almost always preferable to acrylic, especially if there are any physical stresses involved such as in a robot base. Acrylic is notch sensitive and needs the right tools (drill bits, cutters and blades) to properly machine it without inducing stress cracking. It can be rough sized very easily by scoring a straight line from side to side, then snapping it over the edge of a table.
Polycarbonate is a dream to work with in comparison, although you would be wise to get a book from the library or research on the net for good machining recommendations:http://distributionbizwiz.wordpress.com/2007/08/22/machining-forming-polycarbonate/
In some cases, MDF (Medium Density Fiberboard) works as a good project base for a robot . . . it is relatively inexpensive and easy to work with . . . it does not however come in thin sheets.
Baltic birch, marine grade and aviation grade plywood can be easily acquired in small sizes (often 12 x 18”) from many hobby and craft stores. It is available in thicknesses 1/16 – 1” and is completely free of the voids and imperfections normally found in construction grade plywood. It is also made up of thinner plys than construction plywood. It cuts and machines very well and looks good without secondary finishing.
Aluminum sheet (1/16 - 1/4” thick) can be purchased cut to size from many steel suppliers (some operate just like lumberyards). Although more expensive by weight than plastic sheet it has great rigidity and machines well. Perhaps you might want to reserve this for when your robot moves from prototype to permanent device.
Finally do not discount thin sheet metal (approx 1/16”) which can be used to make some very light weight reasonably stiff enclosures that I have also seen used as a robot base. Although more difficult to work with, if you are a university student with access to a shop or are on good terms with your old high-school shop teacher than you can probably get access to some punch and brake equipment. NOTE: that Hammond and some other enclosure manufactures offer inexpensive, simple aluminum chassis with covers that will give you a place to mount your sensitive electronics.http://www.hammondmfg.com/dwg21.htm
Okay, so if you have read this long post this far, I'll give you a freebee! If you are a university student, or work for a company that is involved in any kind of engineering, you might even be able to get a free enclosure that suits your needs. Many manufacturers of plastic enclosures offer a sampling program. Check out their customer service page or dig deep into their site map to find a possible link to their sampling program.