While the merits of handing a 33-year-old a three-year deal are debatable, Barry's worth to the team is certainly not.
Throughout his loan spell last year, the former Manchester City man was integral to Everton's success and he will continue to be so in the immediate future.
Barry not only provided a calming experience, but has—crucially—accelerated the shift to Martinez's possession-based approach. Without him, Everton's transition would have been far more rocky.
Playing the most important role in the team, Barry is regularly seen dropping back between the centre-backs, bringing the ball out of defence and coaxing opponents forward.
His raking diagonals locate Toffee full-backs with space to run and his ability to take minimal touches and find clever angled passes keeps the tempo brisk.
While others grab the headlines, Barry triggers attacks, initiating combinations through densely populated final thirds.
No other Everton player is as comfortable playing this role, which magnifies his importance—especially given the key aspects his duties entail.
Unsurprisingly, he was his club's leading passer in 23 of the 32 Premier League games he played, deployed as the Toffees' link man.
He averaged almost 10 passes per game more than any other team-mate and was one of the most prolific passers in the Premier League. He, more than any player, helped forge Martinez's identity on the pitch.
Out of possession, Barry was just as crucial and something of a ball-retention machine. He won the ball more than any team-mate in the midfield third (131) and third most in the defensive third (77).
He and James McCarthy protected their back four well and formed an effective partnership together. They can progress even further with a second year of chemistry.
One only has to look at Everton's record with and without Barry to appreciate his value.
With the midfielder, the Toffees won 63 percent of their matches (20 of 32 games), equating to an average of 2.1 points per game.
Interestingly enough, maintaining that ratio would have seen the Toffees finish with 80 points from 38 games—good enough for the Champions League.
However, without Barry, only 17 percent of matches were won (one of six) and the Toffees averaged just one point per game while scoring fewer goals and conceding more.
Clearly, it was vital to prolong his time at Goodison Park.
Barry's skill-set is all-too rare among English players and yet pivotal to so many teams, including the Toffees.
Everton's approach, while initially impressive, is still a long way from Martinez's desired end product and Barry's presence will once again aid this development.
Stylistically, he's an ideal fit for this evolving play, and, with so many young players around him, he will continue to benefit the club both on and off the pitch.
It's important Everton not only enjoy his time, but identify potential replacements to learn from him. In a Europa League campaign, there will be chances for young players to absorb his game and attempt to mimic his input. Going forward, this is crucial.
For the next few seasons, however, Martinez has his man. As the Catalan begins plotting a path to the Champions League, a key piece of his puzzle is now in place.