UPDATE 2-UK's Cameron keeps Osborne as finance minister to pursue deficit push

* Three other cabinet members re-appointed (Adds background)

By William Schomberg

LONDON, May 8 (Reuters) - Prime Minister David Cameron reappointed George Osborne as his finance minister on Friday and gave him a bigger cabinet role after Britain's economic recovery helped their party to an unexpectedly big election victory.

Osborne will have a freer hand in his purge of Britain's public finances after the Conservative Party won an outright majority on Thursday, meaning it no longer has to seek consensus with its former Liberal Democrat coalition partners.

As well as keeping Osborne in charge of Britain's economy, Cameron named him first secretary of state, making him the most senior member of cabinet after the prime minister.

Osborne served as Chancellor of the Exchequer throughout Cameron's first five-year term. During that time, he oversaw a recovery of the economy from the damage wrought by the financial crisis.

The improvement was slow at first but gained speed sharply in 2013 and 2014, transforming Osborne, 43, from one of Britain's most unpopular politicians to a contender to succeed Cameron, who has said he will not seek a third term.

Osborne has also halved Britain's large budget deficit, now running at nearly 5 percent of gross domestic product, although he missed his original target to eliminate it altogether by now.

In other cabinet re-appointments on Friday, Cameron announced Theresa May would remain as interior minister while Philip Hammond and Michael Fallon would keep their jobs as foreign and defence ministers, respectively.

As a key Conservative Party strategist, Osborne is expected to play a big role in Britain's attempts to win back some powers from the European Union before holding a referendum before the end of 2017 on whether to remain in the bloc.

He will also help come up with London's response to the rise of the Scottish nationalists, which could involve the devolution of more powers to Scotland. The nationalists won 56 out of 59 seats north of the border.

FREER HAND ON CUTS

Osborne has promised to put the public finances back in the black by 2019 by cutting spending further, including through big reductions in welfare payments.

The Liberal Democrats, who will no longer sit in Cameron's cabinet, said during the election campaign that they had prevented the Conservatives from making some deep welfare cuts.

Osborne has vowed to chop 12 billion pounds from the welfare budget, something the independent Institute for Fiscal Studies has said would require radical change.

Other priorities include a plan to help economic development in northern Britain, which has lagged behind London and England's southeast.

At the same time, Osborne must hope that a nascent recovery in the neighbouring euro zone is not derailed by a deepening of the debt crisis in Greece.

Data released on Friday showed the value of goods sold by Britain to the bloc fell by nearly 9 percent in the first three months of 2015 compared with the same period last year, hurt by the appreciation of the pound against the euro.

Osborne said in 2011 that he would lead a revival of Britain's manufacturers and exporters but that has yet to take off, weighing on Britain's current account deficit which last year was its widest on record. (Additional reporting by Kate Holton, Kylie MacLellan and Michael Holden; Editing by Mark Trevelyan)