Why didn't Noach Daven to save his generation like Avraham Davened for Sodom?
There is a popular idea (I don't know an actual source for it) that Noach was the classic "tzaddik in a fur coat." The metaphor used is that when the house is cold, you can do one of two things: wrap yourself in a warm coat, ensuring your own comfort but ignoring everyone else's; or build a fire (or turn on the heater, etc.) so that everyone benefits. Similarly, then, Noach was concerned about his own righteousness and that of his immediate family, but not so much about everyone else.
[It is true that, as the Gemara (Sanhedrin 108a) and other sources note, that Noach did reprove them throughout the 120 years that he spent building the ark. However, the Lubavitcher Rebbe zt"l explains (Likkutei Sichos, vol. 15, pp. 40-41) that he did so not out of real concern for them, but simply because G-d had commanded him to do so.]
That said, the Zohar (1:67b-68a) cites a discussion between R. Yitzchak and R. Yehudah on this point. R. Yitzchak indeed contrasts Noach unfavorably with Moshe, who prayed for his generation and averted Hashem's wrath against them; R. Yehudah counters that even Moshe was unable to find sufficient merit among his contemporaries, and had to appeal to the merit of the Avos - who of course hadn't yet been born in Noach's time. In another of his talks (Likkutei Sichos, vol. 25, pp. 21-22), the Rebbe notes that R. Yitzchak is not actually faulting Noach for his inaction, since indeed he was laboring under the constraint that R. Yehudah mentions as well as several others described in the Zohar there; he is simply pointing out that we should not use him as a role model.)
From Torah Insights for Shabbat Parshat Noach 5758 by Rabbi Aaron Borow:
Rabbi Ki Tov answers. Noach believed that the people of his time were too far gone. They were beyond salvation. They had no respect for their own humanity or for that of others. Among all those who lived in his time, Noach could not find even a minyan of good people.
Nonetheless, for all their evil, Hashem takes Noach to task for not defending them, for not taking their case.
One lesson of this parshah is that we have no right to give up on people. Even if we believe they are beyond salvation, we must still extend a helping hand and pray to G-d for their salvation.
From Jewish Values in a Changing World by Harav Yehuda Amital zt"l:
Rabbi Yehoshua said: Why didn't Noach pray for mercy on behalf of his generation? He said in his heart: "Perhaps I shall not be spared," as it is written: "For you have I seen righteous before me in this generation" (Bereishit 7:1), i.e., in relation to his generation. Therefore he did not pray for mercy on their behalf.
Noach was unable to pray on behalf of his generation, because such an act would express a feeling of connection and concern. Precisely because Noach was not a greatly righteous man, but only righteous "in relation to his generation," he was concerned that pleading for mercy for them would lead to a blurring of the distinction between him and the rest of his generation.
In order to pray for another person, one must understand his tragedy. To be able to pray for his generation, Noach would have had to understand them, and argue before God that their deeds should be pardoned in consideration of the circumstances that brought them to do what they did. It was this that Noach feared. If you really understand someone, doubts may arise within you - perhaps that other person is really right.
(this is similar to Monica's comment)