No pictures of the fish, so I can't say for sure. Here's one possibility from
http://www.bettatalk.com
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DISEASE: FUNGAL INFECTION GENERAL INFO: If you always add aquarium salt to your bettas water (1 teaspoon of aquarium salt per 2 1/2 Gal of water) and one drop of Aquarisol per gal, your betta will probably never get fungus. It is contagious, but bettas will more than likely recover if treated promptly.<!--mstheme-->[/font]
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SYMPTOMS: Betta has white cottony like patches on its body or head. He may be less active, may have stopped eating, fins may be clumped, color may be pale.<!--mstheme-->[/font]</TD></TR><TR><TD width="100%"><!--mstheme-->[font=Arial, Arial, Helvetica]
TREATMENT: Do a full jar water change. Add Fungus Eliminator by Jungle. These are crystals are dosage should be about 30-40 grains per 1/2 gal. Water should have a nice gold color, not too dark. Do not overmedicate! Change water every third day and add a new dose of same medication. Continue until all fungus has disappeared. Then keep betta jarred for another 2 weeks at least and instead of Fungus Eliminator, just add 1 drop of maroxy or of Methylene Blue in your bettas water to clean any other bacteria/fungus that may still be present. Adding salt to the water would also help healing.<!--mstheme-->[/font]</TD></TR></TBODY></TABLE>
It might also be ich (does it look more like white spots, or is it more of a cottony substance?).
I can't stress enough -- if you keep him in a very small container, you're going to have problems. Spend the $25-$30 on a 3-5 gallon aquarium and you (and he) will be much better off!
Good luck!