According to the very comprehensive A Greek-English Lexicon of the New Testament and Other Early Christian Literature (BDAG) the normal meaning of πέτρα (petra) is "bedrock or massive rock formations". So, most often the writer/speaking will have something like an exposed rock formation in mind, not an individual stone. Matthew 16:18 is classified under this meaning.
A less common meaning is "a piece of rock". That is, what an English speaker typically thinks of when they hear "rock". I suppose this could include tiny rocks (i.e. pebbles), but that certainly is not the normal meaning. And there is no reason to think that πέτρα means pebble it our passage - it would be completely natural to say you are building a church on bedrock, but very strange to say you are building it on a rock/pebble.
So, the answer is πέτρα means rock or bedrock, not pebble.
The poster in the post you link to seems to have been confused, as his claim is not one that is actually made by anyone about that passage, as far as I know. Some people claim Πέτρος (petros), not πέτρα, meant pebble, so he probably mis-remembered what he heard. This claim is pretty sketchy too, though. It is true that in Attic Greek πέτρος usually meant a small rock, but it could also refer to bigger rocks. More importantly, the New Testament was not written in Attic Greek, but rather Koine Greek. In Koine Greek, this specialized use of πέτρος is unattested, and regardless πέτρα, a grammatically feminine word, could not be given as a male name. (πέτρος is the masculine form of πέτρα.) Thus, there is no reason to believe Matthew 16:18 is intended to be sarcastic.