Hair ... or notThe decision on what sort of hair to have is an important one. You can go with wild hair (such as Albert Einstein and Doc Brown from the Back to the Future movies) or bald, the Lex Luthor route.
If going for wild hair, there are inexpensive wigs available in most costume shops. Alternately, you can create your own with a bald cap by gluing fabric hair (from local fabric or craft stores) onto it - perhaps different colored hair. Or, if your hair is long enough, you can use styling gel and unusual hair colorings to get the desired effect.
A bald cap (especially nice for female mad scientists) will also do. To really complete the look, you may want to use some fake skin (try this fake skin recipe, perhaps) to cover up your eyebrows. This will create an effect that you've lost all your hair due to the bizarre potions you've tested on yourself.
A middle approach between the two is to glue chunks of fabric hair onto the bald cap, so that it looks like your hair is falling out in chunks. Again, using hair with strange colors might be useful.
Other HeadgearSome sort of eyewear is generally a good idea. Find an old pair of glasses with bulky frames, perhaps from a thrift store, and pop out the lenses. You might want to decorate them by gluing or taping object to them, like bottlecaps, beads, etc. Tape (duct tape) or Band-Aids can be used to make the glasses look like they've been broken and mended. Goggles are also a nice alternative.
A goattee is a nice addition for a mad scientist. If you can't or don't want to grow your own, you can glue some fur into your chin. Try to frame it into a sharp point, perhaps using a bent paperclip or piece of cardboard as a frame to mount it on.
Lab CoatThe lab coat is, of course, the crucial element of the mad scientist costume. This is what translates the costume from "random weirdo" to "mad scientist." Around Halloween, lab coats are pretty easy to find anywhere costumes are sold. You can also get actual lab coats at medical supply stores, thrift stores, and the like. If you have trouble finding one, you can contact your local hospital to find out where they are sold locally.
Personally, the best lab coat I've ever seen is the Mad Scientist's Union Local #3.14. I didn't buy it online, so I can't attest to this vendor, but the lab coat is very cool.
You can also decorate the labcoat with pins, stickers, stencils, decals, rips, scorch marks, food spills, equations, and the like ... whatever you feel comfortable with based on the cost of the lab coat.
Pants - The Easy PartGenerally, dark pants or a dark skirt will work to finish the outfit.
A goofy pair of shoes, such as bowling shoes, would do well to complete the outfit.
Final AccessoriesA pocket protector (try office supply stores) is a perfect addition to the costume. Fill it up with as many pens and pencils as you can. Throw in a compass, ruler, spiral notepad, and calculator if you can. Heck, carry around an abacus if you can find one.
Another nice accessory would be a beaker filled with strange-colored liquid. Exotic colors of punch (i.e. Kool-Aid) can create this. Add some dry ice so smoke will drift off of it.
Note: If you do have a potion with dry ice in it, do not drink.
A flourescent stick, like what you get at the circus, can be inserted to make it glow ... and is great for stirring up your concoction.
Some Last CommentsThe unbridled wackiness is the best part of the mad scientist costume. Be funny and nuts, and you'll pull it off. Anything you can think of to add to the costume's eccentricity is a plus.
Try to go as cheap as you can, since you may want to make a real mess of the costume to create the right mood. Old pants, tattered lab coats, funny shoes, out of style glasses ... thrift stores are the perfect place to get components for the mad scientist costume.
Mad Scientist Sidekick Costumes