I agree with the side saying that you should do the refereeing for the second, third, etc time if you are asked. In particular I very much agree with no name and Ben in the reasons they cite.
I would add the following. Very often there are only a few people who can truly evaluate a paper's worth. If you are one of those and you decline refereeing the chances of the paper going to someone who has much less perspective on the field is higher. Then some decision will be made, possibly taking much more time and it is not necessarily a better decision than what you would have suggested. If independent editors pick you as the best choice to do the evaluation then you probably are (at least one of) the best to do it.
I suppose that when you decide to suggest rejection of a paper you do that based on the quality of the paper versus the quality of the journal. When you are asked to do the same for a different journal this comparison changes and so it is possible that you would feel that the paper is appropriate for the second journal and suggest that they accept it. You could actually try to influence this in your initial opinion. When you suggest that they reject the paper you could give an example of a journal that you feel is appropriate.
Regarding the statement that submitting to a second journal is like asking a second opinion I am not sure that I agree. On one hand we would all like to ask for a second opinion when a paper of ours is rejected, but fortunately we cannot do that. However, we would still like it to be published, so I think it is more like a second shot then a second opinion.