The former PM told the BBC's Andrew Marr that Labour should change its position if the government delivers a Brexit deal the people do not like.
He urged Labour to hold ministers to account over its pledge to secure a trade deal with the EU.
Labour's Jeremy Corbyn has said the referendum result should be respected.
Mr Blair, who was Labour prime minister between 1997 and 2007, said the government faced negotiations of "unparalleled complexity" if it was to achieve its stated aim of delivering an agreement that replicates as closely as possible Britain's existing trade arrangements with the EU.
While voters had backed Brexit in last year's referendum, he said he believed it was "possible" the public mood would change if it did not result in the promised benefits - and Labour should be ready to capitalise on that.
"A few weeks ago in the House of Commons [Brexit Secretary] David Davis said they were going to deliver a deal with exactly the same benefits we now have from the single market and the customs union - we should hold them to that," he told the BBC.
"If they are going to try and deliver exactly the same benefits as we have now in the single market and customs union, this is an endeavour of unparalleled complexity."
He argued that if control of EU migration is the only reason for Brexit "it is possible - I don't put it higher than that - that people start to think, 'is this really the thing that is going to be important?'"
Mr Blair said in his opinion Labour should say: "We believed in Remain, we still think the best thing is for Britain to be part of the European Union - we acknowledge the people voted against that, we acknowledge therefore the government has a mandate to negotiate Brexit, but we are going to hold them to the test that they have set.
"And if they do not pass that test, then we are going to retain the right to represent the people of this country should their will change, to offer them the option of staying."
Labour should be prepared to alter its stance on Brexit - and if necessary argue that Britain should stay inside the EU, Tony Blair has said.