From the perspective of Kabbala, the Hebrew word for “my bow”, “kashti” is derived from the word “kashe”, meaning “hard”, or “severe“. It is a reference to the attribute of Justice, the “hard” attribute. G‑d said that the rainbow should become the sign of the covenant.
Significantly, the Torah did not say, “The rainbow I have placed in the cloud…,” which would have meant “the well-known rainbow“. Instead, G‑d refers to “My rainbow“, in order to remind us that the cloud He speaks of is not one of the regular clouds that appear in the sky every day. He does not speak of the kind of cloud which discharges rain. G‑d speaks of the Clouds of Glory, such as when we are told, “And behold, the glory of G‑d became manifest in the clouds” (Exodus 16:10)[the prelude to the Israelites receiving meat and bread from heaven at the hands of G‑d].
Anyone who looks deliberately at the rainbow will have diminished eyesight….
The meaning of the words in our verse, “and the bow could be seen in the clouds” is: “the attribute of Justice will [then] become manifest in the cloud.” In fact, we have a verse, “as the appearance of the bow which shall be on a rainy day, so was the appearance of the brilliance all around. That was the appearance of the semblance of the glory of G‑d”, (Ezekiel 1:28) which our sages in interpret as being a warning that anyone who looks deliberately at the rainbow will have diminished eyesight. (Chagigah 16)
You should be enlightened by the following comment in Bereishit Raba (35:3) on the words “I have placed My [rain]bow in the cloud….” “The word “kashti” describes something that is an obstacle for Me.” Is there then such a thing as an “obstacle” for G‑d? Is there anything that He cannot overcome? We must therefore understand the word as derived from the word “kash”, meaning “straw”, the pedicle of a fruit.
Why did the Midrash repeat the same comment twice? Look at the conclusion of the passage! Here G‑d speaks to Noah alone [not to his children] when He says, “this is the sign of the covenant which I have established between Me and all of mankind.” At this point G‑d took Noah into His confidence and revealed a mystical aspect to him that He had not revealed to his children. This is the deeper meaning of the words, “this is the sign [i.e. visible symbol] of the covenant.” (Gen. 9:17)
[Selected with permission from the seven-volume English edition of "The Torah Commentary of Rebbeinu Bachya" by Eliyahu Munk.]