First of all, here's the actual quote from R' Nissim (pg. 273, bottom), with my translation:
"ואמנם חיי הראשונים - כבר כתב בם הר"ם ז"ל פרק מ"ח מהחלק השני, שלא חיו החיים ההם, רק האישים הרמוזים. אמנם כל אדם בדורות ההם חיו הימים הטבעיים הנהוגים בלבד. והיה הזרות הזה באיש ההוא, אם לסבות רבות: למזונו, ולהנהגתו, או דרך פלא, וזרות שקרה לחוזק טבע נפלא. ואם כן, יהיה מהמין קראתיו בשם "פלא". אמנם יש מי שפירש שהכונה בחיים ההם - קיום נמוסיו והנהגותיו הזמן הנזכר, בין בחייו בין אחר מותו. כי אלו, אפשר שהיו אנשי שם, מחקים חוקים ונימוסים. ומנהגים במידותיהם, גם במאכלם ומשתיהם ובמלבושיהם, ואחר הזמן ההוא אפשר שנשתכח הכל ובחרו דרך אחרת. או תאמר, שלא קם כמוהו עד זה הסך מן השנים במעלת ידיעת ההנהגה לבני דורו. ובזמן ההוא קם כמוהו במעלה, נמשך לדעתו וכונתו, ונחה רוחו עליו כאשר נחה רוח אליהו על אלישע. ואם לא ראה הראשון זה שקם אחריו ולא היה בזמנו, אפשר למד מספריו או התבונן בדבריו המקובלים. ועל זה כתוב: "ויולד" - שהוליד בדמותו במעלה, כאלו הוא הוליד האיש ההוא מאשר הוא בעל הדעת ההוא שלמדהו. ואמרו: "ויולד בנים" אחר הסך הרב מן השנים - ענינו וכבר הוליד. אמרו ז"ל בבראשית רבה (ויקרא רבה כא, ט) "בזאת יבא אהרן אל הקדש. הכתוב בשרו שהוא חי ד' מאות ועשר שנים כמנין "בזאת". וכי תעלה על דעתך שאהרן חי ד' מאות ועשר שנה? אלא במקדש ראשון, על ידי ששמשו בו באמונה, נקרא על שמו." ר"ל לפי שנמשכו אחריו לשמש באמונה והלכו בדרכיו, הוא חי כל אותן השנים, דוגמא למה שפירשנו בחיי הראשונים. ולפי זה הפירוש יהיה זה המין השני מהחלק השני."
"Although the lives of the early generations - the R"M z"l already wrote in ch. 48 of the second part, that they didn't live those lives, just the people hinted. Indeed, every man in those generations lived the traditionally natural days only. And this strangeness was in that man, perhpas for many reasons: For his food, and his way, or through some wonder, and this strangeness to strengthen the wonder of nature. And if so, this would be of the type that I call "wonderous" [according to the footnote, he explains this earlier in the book; it refers to uncommon things in nature that happen without the assistance of a (miracle-working) prophet]. Although, there are those who explain that the meaning of those lives - the preserving of his mannerisms and ways for the time mentioned, whether during his life and whether after his death. For these, it's possible that they were great people, making laws and mannerisms. And their ways per their virtues, also in their eating, drinking and clothes, and after said time, it's possible that everything was forgotten and they [the rest of the people] chose a different path. Or should you say, that no one rose up during this total of years with the virtue of knowing how to lead his generation. And in that time someone of his stature rose up, drawn after his thought and direction, and his spirit rested upon him as the spirit of Eliyahu rested upon Elisha. And if the first one wasn't seen by that who rose after him and wasn't in his time, it's possible that he learned from his books or examined the traditional teachings. And on this it is written: "and he begot" - that he begot in the image of his virtue, as though he had given birth to that man, from that of which he is master of the thought of that man who taught him. And what is said: "And he begot children" - after the total of many of the years - meaning, and he already begot, as they said z"l in Beresheet Rabbah [Vaykira Rabbah 21:9]: "Thus only shall Aaron enter the Shrine. The verse is telling him that he lives 410 years, as is counted in "Bezot" [thus]. But would you even consider that Aharon lived 410 years? We must say that in the First Temple, through the fact that they served in faith, it was named after him." Meaning that because they were drawn after him to serve in faith and walked in his ways, he lived all of those years, an example for what we explained about the lives of the early generations. And according to this explanation, it is of the second type from the second part [according to the footnotes, again a reference to an earlier part of the book; in other words, strange things that happened not at the hand of a prophet and one should not understand them in the simple understanding [כפשוטו]].
As I understand it, R' Nissim meant that each of these people created some sort of legacy or dynasty1 that was the equivalent of hundreds of years of life. I didn't see him say that they actually had that many descendants, just that their legacy lived on for that many years. In Noach's case (R' Nissim doesn't refer specifically to his age, not here nor in Parshat Noach), I would say that to some extent, his influence on the world was the equivalent of 600 years prior to the flood.
Now, one may ask: In what way did he influence the world? Nobody listened to him and they all died!
Which brings us to Tanchuma Beresheet 12:
"What do the words in our work and in the toil of our hands (ibid.) imply? Prior to Noah’s birth, men did not reap what they sowed. They would sow wheat and reap thorns and thistles, but when Noah was born, the world reverted to normal: Wheat was sown and wheat was reaped; barley was sown and barley was reaped. Furthermore, prior to Noah’s birth, men performed all their labor by hand, as it is written: And in the toil of our hands, but after Noah was born, plows, scythes, axes, and other implements were introduced."
According to the midrash, Noach invented plows and other farming tools. It seems that according to R' Nissim, this was so influential that it was the equivalent of 600 years of life.
And after the flood, he continued to influence the world for a few more centuries, being now the father of humanity. It seems the his influence completely died out 349 years after the end of the flood (plus one year in-between, total - 950).
1 This interpretation works surprisingly well with the view among some secular scholars and one rabbi (that I know of) that say that the ten generations from Adam to Noach (through Shet) were kings of ancient Babylonian cities. For more info, see here.
