汉 (traditional 漢) began as the name of a river — the water radical 氵 gives it away. The Han River lent its name to the Han dynasty (206 BC–220 AD), and the dynasty was so defining that the people took its name forever.
汉 names the civilization: 汉字 (Chinese characters — the very things you're learning), 汉语 (the Chinese language), 汉人 (the Han people, 92% of China). One river became a dynasty, the dynasty became a people, and the people named their writing after it. When you study hanzi, you're literally studying 'Han characters' — a 2,000-year-old brand. Also: 好汉 means a brave man, a hero.
Water 氵 from the Han River — the river that named a people: HAN.
The Han dynasty was contemporary with Rome — and while Latin script changed beyond recognition, 汉字 are still recognizably the same system.