It wasn't a British move, it was an American project and Blair was Bush's lapdog. We eased a bit of the financial burden for them and offered some of our troops for fodder.
I'm talking about why Blair probably backed it. The US reasons are another whole ball game.
Blair's power delusion started during Clinton's Lewinsky impeachment . If you remember, he provided Clinton a character reference in front of US/UK TV, and in doing so, implanted the seed of his global importance into his mind. Blair changed dramatically after this, seeking to become a global statesman, despite still being very early into his PM tenure.
I always found it odd how Blair immediately schmoozed upto Bush, despite them being at the opposite ends of the political spectrum. As soon as 9/11 happened, Blair jumped at the chance to be involved in war to acquire global importance, despite 9/11 having nothing to do with him. The wars in Afghanistan and Iraq would certainly have always happened, but there was zero reason for UK to have gotten involved, to the extent that we did. We have to remember that no other NATO nation volunteered its soldiers in the 'coalition of the willing'. I don't think Blair was a lapdog; he was a willing and passionate advocate for Bush agenda.
It's also farcical with hindsight to talk about any moral justification. There was none. The Iraq war was simply the end of the Bush family feud with Sadam Hussien and oil. I wonder how many young people learning about Iraq war for first time know Saddam had nothing to do with 9/11, the false links of WMD, the sexed up dossiers and the suspicious death of Dr Kelly, Abu Graib etc. It's a very shady era of British history where many crimes were committed by Government and Army people. As I wrote in my earlier post, it's unforgivable, and rightly overshadows any positive achievements Blair might have achieved.
Blairs bromance with Bush was purely on his own narcissist desires to become a playa of global importance (cf: his infamous ‘Blair Doctrine’ mentioned earlier by
@nickm). The UK is still paying this price, with terrorist acts like the one over the weekend, let alone Iraq and the region.