Former international Brady, who was part of Giovanni Trapattoni's coaching staff for the first two years of his reign, is convinced the man who took the Republic to the 2002 World Cup finals in the Far East is the one for the job once again.
Ipswich boss McCarthy famously emerged from his Saipan spat with then Ireland captain Roy Keane with his reputation enhanced as his team made light of a major loss which divided fans back at home.
Brady told RTE Radio: "For me it would be Mick McCarthy. I don't know what his contractual situation is at Ipswich, but what I do know is that he's demonstrated before that he's capable of doing the job.
"He has broad shoulders, he had to put up with an awful lot of messing when Saipan happened and people turned against him.
"Most of the media turned against him, public opinion turned against him because of how popular Roy Keane was. But the man got on with it.
"He's demonstrated since then he's a very, very capable manager. I don't know if it's possible to get him, but he would be number one on my list."
The Football Association of Ireland is currently engaged in the process of identifying potential successors to Trapattoni, who parted company with Ireland last month, with McCarthy, Martin O'Neill and Keane among the leading contenders.