Rashid, a 37-year old professor of media at Kuwait University, replaces Muhammad Abu al-Hassan who quit on 3 January under pressure from Islamist MPs who filed a motion to quiz him over morality issues.
The new minister completes the oil-rich emirate's 16-member cabinet headed since July 2003 by liberal-leaning Sabah al-Ahmad al-Jabir Al-Sabah, half-brother of the amir.
Rashid's appointment comes before an anticipated crucial vote in parliament over a government-proposed bill to enfranchise Kuwaiti women.
Under Kuwaiti law, ministers, though unelected, have the right to vote in parliament.
But the appointment of Rashid - a Sunni - leaves the Shia minority without representation in the cabinet for the first time in several decades.