I often carry a "Platypus" .5 ml collapsble water bag because it is much lighter and less bulky than a bottle and can go in a purse side pocket. I empty it before going through security and refill from a water fountain, faucet or if necessary from bottled water. I have gotten some funny looks when drinking from it because it looks like an IV bag. It comes in different sizes and has acessories including filtration kits. I bought mine at a sportswear store.
http://www.platy.com

Some security personnel must get extra points for over-caution. About 30 years ago when hijaking of planes was common one of my boys put his day-glo green "brass knuckles" water gun in my purse. The "knuckles" went over the fingers. The reservoir which was hidden in the fist resembled a hand grenade so security would not let it go on the plane nor would they allow a stewardess to put it away until after the flight. They could plainly see that it was a toy but we had to get a box and put it through as checked luggage.
I carry a small spray bottle of water in my purse to mist my face for cooling and setting makeup, to spay on my "coiffure" when I don't have time to set it, and as part of my on-the-go watercolor kit. I once had to do the TSA Hokey Pokey because it showed up on the xray of my purse. If I had been asked I would have shown that it was empty.

Malka