What is the ruling on a brother supplicating for his brother to die as a drowned martyr, justifying this by saying that drowning is easier than other forms of martyrdom like dying by burning or stabbing?

Your brother should not make this supplication or ask Allah for such a death, for the Prophet, peace and blessings be upon him, used to seek refuge from a bad death. In fact, it is authentically reported that he used to seek refuge from drowning. Al-Nasa'i and others narrated from Abu al-Yasar, from the Prophet, peace and blessings be upon him: "O Allah, I seek refuge in You from falling from a height, being crushed beneath a falling object, drowning, and being burned. And I seek refuge in You from Satan striking me down at the time of death. And I seek refuge in You from dying in Your cause while retreating. And I seek refuge in You from dying from a poisonous sting."

As for asking for martyrdom, that is a good matter. It is authentically reported from him, peace and blessings be upon him, that whoever sincerely asks Allah for martyrdom, Allah will grant him the stations of the martyrs, even if he dies in his bed. In the Sahih (definition), it is reported that Umar, may Allah be pleased with him, asked Allah for martyrdom in His cause. If your brother desires good, let him confine himself to what has been reported, and let him ask Allah for martyrdom in His cause alone.