Was the rainbow just created at the time of the flood? Ibn Ezra (on 9:13) argues yes, based on the word natati (9:13). Radak (on 9:13) explains that according to this approach after the flood G-d made the sun stronger that it would cause a rainbow. On the other hand, the Mishnah (Pirkei Avot 5:9) records that the rainbow was created during the twilight of the eve of the first Shabbat. Similarly, Rabbenu Saadiah Gaon (on 9:13) and the Ramban (on 9:12) argue that the rainbow always existed and only after the flood was its appearance to be taken as an ot, sign, of the covenant. (Does Radak suggest an in-between approach that it existed before the flood but did not exist during the year of the flood?) The text here supports the Ibn Ezra, but it is hard to argue that the physical properties of the sun and water changed from before the flood to after the flood.
Another question is the purpose of the rainbow. It is commonly understood that the rainbow informs a person that really G-d should have destroyed the world but G-d did not do so since G-d had promised not to. With this idea, the rainbow is for people to realize that they are misbehaving. However, 9:15 states that rainbow was for G-d to remember.
The language of G-d remembering is also not clear since we believe that G-d does not forget. Instead, the word remembering in reference to G-d is traditionally understood as a way of expressing the idea, in human terms that G-d is acting after ignoring something but in this case what was G-d ignoring?
Cassuto (1964, pp. 138,139) notes two other difficulties with the sign of the rainbow. One, 9:13,14,16, all record that the keshet was to be in the clouds, but rainbows are in the sky not in the clouds. Two, how can the rainbow remind G-d to not wipe out the world, if the rainbow appears after the storm is over?
While Cassuto suggests answers to his questions, I doubt the word keshet here means rainbow. I believe that after the flood, G-d put in the clouds (atmosphere?) a mechanism to stop the rains at some point, a type of automatic shut-off system. The language of G-d remembering is in reference to the system that the system that G-d set up would act to stop the world from being flooded. This action would occur after the rains had begun, which from the system's point of view would be ignored until the rains reached a certain "danger" point, and then the new system would stop or slow down the rains.
With this idea, the word, ot, sign, does not mean a sign that people will see on a regular basis but it is a sign that the covenant is being fulfilled and is not mere words. Thus, G-d established the system to stop the rains from flooding the world, and this indicates that the promise of the covenant of 9:8-11 is being fulfilled. This would be similar to the ot by circumcision, 17:11, that it is not for a person to always look at this sign, but when parents circumcise their sons, then they have demonstrated their commitment to the covenant. This demonstration is the ot of the covenant.
Bibliography:
Cassuto, Umberto (1883-1951), 1964, A commentary on the book of Genesis, part two: From Noah to Abraham, Jerusalem: Magnes Press.