in the original sentence, 'saying' is a present participle. Adding 'were' changes this to the past continuous. Note: - You can't add the 'that' in brackets, it doesn't fit here. - Adding 'were' changes the meaning; without 'were' it simply means you have a memory of them saying this at some time in the past. With 'were' only works if you are describing a specific moment during the time you heard them saying this, e.g. "I remember they were saying he was a really nice kid when he shot them". - You can slightly rewrite the sentence to make 'saying' into a gerund: "I remember their saying he was a really nice kid". This is correct, and has the same meaning as the original, but sounds rather pedantic.

Gerunds and present participles look the same but function differently. I like cake. Cake is delicious. (cake = noun) I like swimming. Swimming is fun. (swimming = gerund which functions as a noun) After swimming, I took a nap. (swimming = present participle which functions as a verb)