No they should not.
Ichiro is 41 years old.
Ichiro is not a very good baseball player anymore.
The Mariners have a lot of not very good baseball players.
The Mariners' team wRC+ in June is 74.
Ichiro's wRC+ in 2015 is 75.
Mariner outfielders are bad.
Mariner outfielders with a higher wRC+ than Ichiro: Seth Smith, Nelson Cruz, Austin Jackson, Justin Ruggiano, and Mark Trumbo.
Ichiro is no longer the plus defender in the outfield he was for so many years.
Dustin Ackley, who many fans never want to see take another at-bat in a Mariners uniform, has been worth -0.6 fWAR in 2015. Ichiro has been worth -0.5.
The Mariners would have to actually give up something to acquire Ichiro, who is now bad.