Some of the arguments I've heard on the "No" side basically come down to that the person who wins under AV could be different to the person who would win under the current system, which is kind of the point (the wording was more that the person who wins might be the person in 2nd or 3rd place, but the concept of 2nd and 3rd place being based on 1st choices isn't how AV works).
You're right in that that argument is not a good one, but there are other arguments that have been bandied around:
-More hung parliaments. Very few people are fans of the current conservative/lib dem coalition, yet with AV we'd see more governments like it.
-Smaller parties fare better, but all smaller parties fare better, including extreme ones like the BNP.
--This is compounded by the fact that, under FPTP, lots of people whose first choice is a smaller party don't vote for them because they don't have a chance, and instead vote for their favourite of the candidates who have a chance. Under AV, this would change.
-There are very few other countries (3?) that use AV, presumeably for a reason. Furthermore, I have heard (though I in now way present this as an absolute fact about all australians) that a large number of australians, who use AV, do not like it.
-Some people's vote will count for more than others. There will be some people who only have their first choice count, and others who have their first, second, third, etc considered.
-As simple as our current system (say who you would most like) results in a lot of people not understanding it (there are tales of people who think you have to 'vote them off'). The new system has a lot more ways that peole can get it wrong (e.g. do you give your favourite candidate the biggest number or the lowest one).