The Sky Sports pundit also thinks Stoke City should follow suit.
Indeed, according to the Sky Sports pundit, neither of the trio are in danger of getting relegated from the Premier League - but nor does Merson rate their chances of qualifying for next season's Champions League.
"If they take every game seriously they have a chance of winning it and it’s been a long time since one of them won any silverware.
"Everton and Stoke have both taken the Capital One Cup seriously this season and they both made it to the semi-finals. They should do the same here.
"Along with West Ham, they are not getting into the Champions League and they are not getting relegated. So they should put everything into the cups."
For clubs with a slim chance of being dragged into a battle against relegation, being gung-ho in the FA Cup would be a risky approach.
But Merson's comments certainly make sense for all three of West Ham, Everton and Stoke. Of course, if Everton or Stoke reach the League Cup final, there is less pressure on them in this tournament.
And yet - imagine how good their season could be if they went very far in both competitions?