This is indeed an idiom. More precisely, it's a corruption of one. It comes from the phrase "drop of a hat":
Bob is always angry. He's willing to start a fight at the drop of a hat.
There's a question at ELU relating to the origin of this phrase. The general opinion is that it comes from the (supposed) Old West practice of an arbitrator dropping a hat to the ground to signal the beginning of a fight.
*How does this relate to the phrase in question? Consider message boards or chat channels. In these communication media, when someone posts something radically opinionated, it tends to start a strongly heated discussion, or even a fight. Someone "dropping an opinion" on the channel, therefore, is someone attempting to instigate.
In the given context, of course, the phrase seems to be used positively, in the same sense as:
Bob is a great guy. He's always willing to help out at the drop of a hat.
"Drop of an opinion" is not standard English, so be aware that use of the phrase could lead to confusion. "Drop of a hat" would be clearly understood by most native English speakers.
**This paragraph is original research*.