One of the themes in "The Red Convertible" is brotherhood/family. Lyman and Henry share this bond and the convertible gives them an opportunity to become closer. When Henry goes off to war, Lyman writes him letters and gives him updates on the car, as if it is their child, something that unites them and uniquely symbolizes their bond:
I wrote him back several times, even though I didn’t know if those letters would get through. I kept him informed all about the car. Most of the time I had it up on blocks in the yard or half taken apart, because that long trip did a hard job on it under the hood.
Just as the long trip wore down the car, the war wore Henry down. Experiencing the trauma of war (a second theme), Henry becomes a different person. He's suffering from post-traumatic stress disorder. The convertible is the only thing that gets Henry out of his distressed state because it is busy work and at some level, the car reestablishes a sense of normalcy and maybe even the brotherhood Henry has with Lyman.