Facebook will no longer allow you to keep your name hidden from searches within Facebook, taking out the "who can look up your Timeline by name" option from the site's security settings. If you have the feature enabled, sometime in the coming weeks you'll see a warning notice letting you know you're now searchable.
Changes last year in how Facebook shared information meant that simply hiding your name from search really didn't do much to keep you from being found. A determined individual could find you by looking at your friends' profiles, or by locating you in photos. It also made Facebook search appear wonky, because someone could be looking at a picture of you but still not find you in search.
How to Stay (Mostly) Hidden
What Facebook would like you to do, and what is arguably safer anyway, is to use more fine-grained controls for who sees what you post. Facebook has some quick links to some of these controls in the upper right of each page, accessible by clicking the padlock icon. Here you can quickly adjust who sees future posts, add people you'd like to block, and change message filtering options.
More options are available by clicking Settings. To stay below the radar, you're going to want to click on Privacy, and review the options there. Note that you can retroactively change permissions on who can see older posts, and hide your profile from search engines like Google and Bing.
If you're concerned about people tagging you, or seeing pictures and posts that you're tagged in, click the Timeline and Tagging link. The settings to control who can see your posts, who can see posts that you're tagged in, and the option to vet posts you're tagged in are critical for keeping a low profile.
Remember that when you make a post on Facebook, you have the option to only allow specific people or groups to see your post. You can also exclude specific people from seeing the post.
Since you can no longer dodge Facebook searches, blocking individuals is probably the best way to keep prying eyes away from your profile. The blocking section of Facebook's settings has a number of options, but restricted lists seems quite powerful. Facebook explains it thusly: "When you add friends to your Restricted list they can only see the information and posts that you make public." This means you can be friends with someone, or friends of friends, but that person can only see public posts. Perfect for stealth-unfriending.
Remember that there are other security issues to be aware of when it comes to Facebook. Phishing messages can proliferate through friends' feeds, and everyone should activate two-factor login for better security.
Several desktop security suites now include "privacy checking" options that will spider through your social media settings and suggest changes. Trend Micro Titanium Internet Security 2014 was effective in our testing at PC Mag labs. There are also web services, like mypermissions.org and secure.me that will check different security aspects of your Facebook presence.
In the years since Facebook debuted, I've noticed people being consistently surprised when they find out how Facebook stores and shares your information. I'd encourage all of you to sit down on your Facebook account and browse through the security settings. If you're still not comfortable after doing that, seriously consider changing how you use Facebook. It can only share as much as you give it, and sometimes the best solution is to just not feed the beast.