The New Augustus
Following Stilicho's death, Honorius' health was declining. Some historians point to his close relationship with Stilicho and the sadness and sorrow he felt upon his death as the reasons for his ailing help. The agreed upon consensus is he suffered from Edema, which lead to his death.
Regardless of how and why his death began to decline, it is clear Honorius recognized that he was not going to live much longer and began thinking about succession. Without a male son to elevate as his new heir, Honorius turned to Constantius, who had effectively taken over Stilicho's position as magister militum upon his death. In 422, he adopted Constantius and elevated him to co-emperor with himself, making it clear that it would be Constantius who would succeed him on the throne.
Thee eastern court vehemently opposed this move, with Theodosius claiming Honorius was nothing but Constantius's puppet, and had elevated him against his will. He would later point to Honorius's death as evidence of poisoning by Constantius to have the throne completely for himself. These accusations were outlandish and untrue, so much so that even the Senate in Rome found them quite ridiculous.
The elevation of Constantius to co-emperor, and his eventual rise as the sole ruler of the west, would have far-reaching consequences. The had effectively escaped what could have been a dangerous whole of being ruled by "barbarian" rulers with no allegiance to the empire. Honorius was determined to make sure the next emperor would be no puppet, and instead a strong and capable leader. Constantius was just the type of man the Roman Empire would need, a strong military leader, who commanded the fervent loyalty of his troops. There would be no usurper to rise up in the west upon Honorius's death, and when Theodosius came marching west to take him down, Constantius would be well prepared.