According to 说文解字,
漢 is taking as its pictophonetic part.
1. What does mean?
This is the ancient inscriptions (甲骨文 and 金文) of it (from 字源谈趣).
Its original meaning is to fire someone (as a penalty), who is tied up and crying to the sky. So it's used for the meaning of
disaster, and then
嘆 were created based on it.
漢 took this character as its pictophonetic part?
1st perspective, from 字源谈趣,
[“漢” is taking “氵”（water） as its ideographical part, taking “漢”（getting rid of 氵） as its pictophonetic and ideographical part. It indicates the largest nationality residing at
漢水 struggled to grew up (into the han nationality).]
2nd perspective, from 象形字典,
[In ancient times
漢水 served as a place of banishment for criminals.]