Yes, it was well known. This was because Feroze Ghandy, her husband, was a well known freedom fighter. (He changed it to 'Gandhi' after joining the freedom movement) Moreover, their marriage even caused a little sensation because Jawaharlal Nehru was opposed to it and even asked Mahatma Gandhiji to persuade them to not marry. After Independence, Feroze Gandhi was well known for his fight against corruption, exposing scams while also criticizing the government of his father in law. He was also elected twice to the Parliament. Thus he was a well known person in his own right.

Moreover, Indira Gandhi's tryst with power began because she was the daughter of Nehru. It was this fact which made her elevation to the post of Prime Minister possible. She was a female in a very male dominated time and politically not very experienced, thus her political rivals believed they could set her up as a leader because of her lineage and then control her. She was projected as Nehru's daughter and successor for this purpose.

Edited for sources- India Since Independence by Bipin Chandra (for the "her political rivals believed they could set her up as a leader because of her lineage and then control her"). Also India after Gandhi by Ramchandra Guha (for "his fight against corruption, exposing scams")

Edited to give a more in depth answer- Indira Gandhi was the only daughter of Jawaharlal Nehru. This fact alone makes her noteworthy to the general Indian public, given her fathers significance. She was also known at the time in her own right. She had been elected Congress President in 1959, the first woman to hold the post since Nellie Sen Gupta in 1933. Moreover she also helped her father in the elections, and accompanied him to his election rallies. Thus the fact that she was related to Nehru was visible during these election rallies. Apart from this she was also the Minister of Information and Broadcasting in the Government succeeding her Father and preceding her own.

As to the name 'Gandhi', it is not a very common surname. It is mostly limited to Gujarat and Punjab. Parsis wrote the name as "Ghandy" (such as Jeahangir Ghandy). Another variation is Gandhy (Kekoo Gandhy). This is not to say that it is rare in any way, for there have been famous people with the surname Gandhi.

Thus while for someone not familiar with India, or for someone completely aloof from the political scenario, there could be doubts and misconceptions about the name, one must remember that at the time, Nehru was a living legend, thus his only child, who was also politically active and accompanied him to his rallies, would be known to the people to be related to him and not the Mahatma.

Related questions: How well informed were average Indians of politics during the early times of India? Also, how invested in it were they?