You will need up to 12 to complete the portal. There may be some eyes already in the frame. When you're throwing the eyes to locate the stronghold, theres also a chance of them being destroyed. I generally take 15 eyes of ender on my first trip to a stronghold.
I believe that 14 is the absolute minimum you need to find your stronghold and fill the End Portal frame. The frame needs a maximum of 12 Ender Eyes to open it and if you can do it right you only need a minimum of 2 eyes to find the stronghold. If you are not familiar with him, The Minecraft Muse is doing an LP where he is traveling to the end using as little eyes as possibe. In a couple of the earlier episodes he details how to triangulate the location of the stronghold by using a map and two eyes of ender.
The portal is activated using 12 ender eyes, and it is found when you throw the ender eye in the air. Sometimes, however, the eye may disintegrate in midair, or return to the end, and it cannot be picked up when that happens, so be sure to only use the amount necessary to find the stronghold. Bring 20, just to be sure. And sometimes the end portal frames are already filled with an ender eye before you get there, so 12 is just if they are all empty.