I am getting pebbles clogging up the cold water side of my bathroom faucet, whats up with that?
you may be getting limescale, that will form stones, but its unusual. You will need a water softener if this is the answer. Try crushing one, it should be fairly soft
If you have just had work done on the plumbing, you could be seeing sediment from the interior of the pipes that was dislodged during the work. I find it odd that you are seeing this only on the cold water side (assuming you mean you don't have a mixer, but separate taps?). Don't install a water softener, as softened water is very bad for you, your children, you pets, and your plants. See what a plumber has to say about it.
Could be calcium. You can flush it yourself but you'll likely need two people for the job. Turn off the water under the sink at the shutoff valve. Remove the faucet handle, then the valve stem(usually some kind of nut holds it on),then the seat and spring. Hold a cup upsidedown over the now open resivoir and have the second person turn the shut off valve ever so gently so the water flows at a manageable rate, hits the cup and goes down the basin. Let it run for a few minutes to clear the line. Put your faucet back together. Good Luck!