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.
Current skipper Robbie Keane revealed on Monday he would be happy with any of the three, but admitted expectations for the national team were perhaps too high, and Brady agrees with him.
He said: "I think the level of expectation in this country for the soccer team to achieve qualification either in the World Cup or European Championships is a little bit exaggerated, so you're going to need a manager, as Robbie says, who has courage and believes in what he's doing and basically ignores opinions going on around him to try and achieve results which are difficult to achieve.
"I'll say it again, the Irish team we have at the moment is lacking in star quality.
"Robbie Keane himself, Richard Dunne are coming to end of their careers. These are players who were big, big players for the Irish team.
"The lads replacing them are not as good. That's the reality of the situation. I think Robbie is talking about a man with broad shoulders. That's what we need."
Indeed, Brady believes early favourite O'Neill may have been put off the job for that exact reason.
He said: "We all expected that Martin O'Neill would be the next manager, but it hasn't happened, so I would think that rules Martin out.
"He probably doesn't fancy the job for the reasons I've already stated. The expectation of the situation is too exaggerated."