He is the most senior figure to join the race, with Glasgow Cathcart MSP James Dornan and Julie Hepburn, who is not an elected politician, also in the running to replace Angus Robertson.
Mr Brown previously stood for the role four years ago but lost out to MP Stewart Hosie.
The former Royal Marine said getting the party ready to fight and win elections and any future referendums would be his focus in the role.
The deputy leadership position became available when Mr Robertson stepped down in February.
Mr Brown said he had been approached by a number of party members and elected members asking him to stand.
“I’ve taken the advice of senior colleagues past and present, people I like and respect, and I’m pleased to be putting my name forward,” he told a national newspaper.
“There’s a serious job for the depute leader in supporting Nicola and getting the party in shape to face the future.”
He added: “The SNP is also the biggest party in Scotland by a long way and we need to get all of our members ready to fight and win elections, and to be ready for any future referendums - getting the party ready and giving the members the tools they need will be my focus.”
The vacancy has arisen after Mr Robertson stood down last year after having lost his Moray seat in the general election which saw the SNP lose 21 seats north of the Border.
Mr Brown served in the Falklands before entering politics and was first elected as an MSP in 2007.
He has held a number of ministerial roles since 2009.
Mr Dornan, who chairs Holyrood’s education committee, has rejected suggestions that the party’s push for a second referendum was behind its reverse in last year’s Westminster vote and wants to push for a second vote on leaving the UK.
Ms Hepburn, a party worker, has said she wanted to take on the role to co-ordinate policy development and push through internal reforms.
The Glasgow Provan MSP Ivan McKee has also indicated that he may stand for the post.
Westminster leader Ian Blackford and several high-profile MPs including Pete Wishart, Johanna Cherry and Tommy Sheppard have ruled out putting their names forward.