This is a false claim fabricated by Ibn Taymiyyah, the godfather of Wahhabism and most of the terrorist groups. He made this claim in his book Minhaj al-Sunnah , volume 1, page 35. The term Rafidhah was used long before the time of Zayd ibn Ali ( AS).
Rafidah is plural of Rafidi from Rafadha ( رفض) which means in Arabic language ( to reject ). It was used for any person or group who rejects something. The term Rafidhah is commonly used for the followers of Ahlul Bayt who rejected the Saqeefa rulers.
1. The enemies of Imam Ali (AS) who rejected his leadership and fought against him in the battle of The Camel, were called Rafidha of Basra. This is been recorded in a letter from Mu'awiyah Ibn Sufiyan to Amr Ibn al-Aass. ( Nassr bin Muzahim, Waq'at Siffeen, P 34). This goes to the time after the battle of The Camel when Zayd ibn Ali was not born.
2. When Al-Farazdaq , the famous poem, praised Imam Ali Zayn al-Aabideen Al-Sajjad (AS) in front of the Umayyad tyrant, Abdul Malik Bin Marwan asked him: Are you a Rafidhi? Al-Farazdaq replied him: If loving The Progeny of the Prophet is Rafdh, I am a Rafidhi. ( Amaali al-Murtadha, V. 1, P. 68). This incident took place in Makkah when Zayd was either not born yet or very little child.
3.Imam Muhammad Al-Baqir (AS) who was martyred eight years before the revolt of Zayd, mentioned Rafidhah. Abul Jaroud narrated that a man told Imam Muhammad Al-Baqir: A person called us with a name. Imam asked him: What is the name? He replied: He called us Rafidhah. Imam al-Baqir pointed to him chest and said: I am also from the Rafidhah and Rafidhah are from me. ( Bihaar al-Anwaar , vol 65, page 97).
4. The term Rafidhah was mentioned in many narrations from Imam Muhammad Al-Baqir (AS) and Imam Jafar Al-Sadiq (AS) as a title for seventy believers who left Firaon and joined Prophet Muses (AS) and called Rafidhah because they rejected Firaon.( Al-Mahaasin by al-Burqi, page 119).
5. In many Hadeeths from Ahlul Bayt (AS) about people calling the followers if Ahlul Bayt as Rafidhah, the Imams said: Blessed name for you, because you rejected the wrong and believed in the Truth, while your enemies have rejected the Truth and believed in the wrong.(Tafseer Furat al-Koofi, page 376 and 377).
6. Al-Shafi'ee who was a well known Sunni Imam has mentioned in his poetry : If loving Ahlul Bayt is Rafdh, then let all humans and Jinns witness that I'm a Rafidhi.