The former Magpies midfielder says the club must allow Benitez to run the club "from top to bottom".
McClaren was sacked on Friday after the damaging home defeat to Bournemouth last weekend left the club second-bottom of the Premier League table with just 10 matches of the season remaining.
Lee, who played 379 times in nearly a decade at St. James' Park, insists that the Newcastle board must allow Benitez to run the club in more areas than just the training ground and the dugout if the gamble in appointing him is to pay off.
The 50-year-old told Sky Sports: "I think the club from top to bottom needs to change. They need some people like Rafa Benitez who are football people, who know how to run the club right from the youth team to the first team and the scouting system.
"It's no good Newcastle putting in place these managers or coaches and then having a policy where there hands are tied on what players they bring in, how old they are, how much they pay for them. I'm sure that Benitez will want to agree that he has a big say on transfers which I don't believe other head coaches have had in the past.
"I think from now on Newcastle have got to get somebody in charge - Benitez will be great - who runs the club from top to bottom and says even if I'm gone in two or three years, this is the way the club should be run."
The Magpies have become known for directing much of their transfer policy from the boardroom rather than allowing the manager to have full control, a system which is becoming increasingly favoured by some Premier League clubs.
However, Benitez has succeeded most when he has been allowed to act as something of an auteur at a club, and in order for he and Newcastle to be a harmonious match, the boardroom may have to hand the Spaniard more control than they allowed McClaren.
The North East club are currently second-bottom of the Premier League table, one point from safety with a game in hand over their fellow strugglers.