No Sword has an interesting post on the etymology of the Japanese arigatou. Matt adds that the notion the word is from the Portuguese word obrigado is “wrong, wrong, wrong.”

Arigatou is the Western Japanese way of saying arigataku, the adverbial form (more or less) of arigatai. The basic rule is replacing ku with u, but in this case that produced au, which becomes ou († a lengthened o) through the magic of sound change. This also happened to omedetou († (o)medetai) and ohayou († (o)hayai). (It is not a coincidence that despite the Tokyo version getting “Standard Japanese” status, the standard politeness terms were imported from the old Imperial capital!)

source: Sorry, Portugal, No Sword, February 3, 2006