It was a weirdly funny moment, but I could not understand it. In The Big Lebowski, when the Dude's White Russian cocktail is drugged, he sees a weird dream involving Bowling, dancing, Julian Moore's character and for a shocking moment appears - one Saddam lookalike. What's the significance here?
The movie takes place during the first Gulf War. As the stranger says at the beginning:
Now this here story I'm about to unfold took place back in the early nineties, just about the time of our conflict with Saddam and the Iraqis. I only mention it 'cause sometimes there's a man [...] for his time and place.
You can also see a Bush speech against Saddam on TV in the background of one of the scenes and Walter also comments at some point:
Look at our current situation with that camel f&*%er over in Iraq. Pacifism is not something to hide behind.
As the dream sequence strings together various elements from the Dude's life, an appearance by Saddam isn't that surprising. It was on everyone's mind back then. [You could, however, hypothesize some deeper connection to the Gulf War: That the Dude's pacifistic approach (or 'abiding') is contrasted with Bush's MO in Iraq, represented by war vet Walter's views and tactics during the film.]
In a footnote of his essay The Big Lebowski: The First Gulf War and Masculinity, Daniel Keyes reports on a comment of George Bush senior’s pronunciation of “Saddam” in The Demonic Comedy: Some Detours in the Baghdad of Saddam Hussein by Paul William Roberts:
Roberts notes that Saddam (with the stress on the second syllable) translates somewhat literally to “learned one.” On the other hand, Saddam (with the stress on the first syllable and nasal “a” as George Bush used) translates to “shoe-shine boy.” Bush was as conscious of this as he was of the way his singular pronunciation sounded to non-Arabic speakers: Sodom Hussein (Roberts, The Demonic Comedy, 125). Ironically, in the dream sequences of Lebowski, Saddam is featured as a bowling alley attendant who polishes shoes.
The book in question is from 1997, and it is plausible that news of the meaning of Bush’s pronunciation have been around much earlier – all before the movie, which is from 1998. While there are several interpretations of Bush’s pronunciation of “Saddam” to be found on the web, they all evolve around shoes.
As already noted, the events of the movie happen during George Bush’s Gulf War and a TV speech of his is featured briefly in the background. So Saddam cleaning shoes may be a result of the Dude’s brain processing some report about Bush’s mispronunciation. It could also be that the Dude speaks Arabic sufficiently well to process Bush’s pronunciation directly, though I consider this less fitting.