Can internet searches cost you your clearance?

Big Brother is always watching and listening, and they have many tools. Sniffer programs key on searches, phrases and keywords. Unless he triggers monitoring and is not breaking the law he probably has nothing to worry about. If he googles "Build a Nuke" it might spark interest. If he starts trying to acquire these items and leaves a paper trail with credit cards, he may have an issue. Most of those guys use Dark Web, but Feds are on there also. Don't break the law or show intent to and you'll be fine.

Not likely, unless he is doing his googling from a government computer. Actually, most of the really cool stuff is blocked on the .mil. As a cyber instructor, there are things I have to research which can't be done from the .mil so I have to do them from home.

Depends on the subject matter. A civilian can lose a job because of the internet such as face book history and so can a Soldier get himself in trouble. Researching how to make weapons also depend on what you want to make. Bombs are considered weapons of mass destruction. I believe there is monitoring of selected site such as that and other types of weapon modifications. A good example is converting weapons to fully automatic fire. The internet is a dangerious research facility to stupid people.

Looking for the old Anarchist Cookbook is a good way to get put on a watch list. It may not have an immediate impact, but if they check said watch list during the next re-investigation itll make for an interesting conversation.

He can search away. If he's searching on a government computer it might raise some questions, since it's monitored. Short of the FBI/ATF arriving on his doorstep, searches shouldn't trigger any issues for a normal secret clearance, which is what a normal support officer would have at an SFG.

Three things:
1 - all searches made on a DOD computer are monitored. If a pattern emerges that tends to bring your friend's "reliability" into question, the S-2 may get involved and that can, in certain circumstances mess with a clearance.
2 - searches done on a home computer hooked up to a private ISP should be protected by the 4th amendment so long as he doesn't break any laws.
3 - if Snowden taught us anything (other than the fact that he's a treasonous turd) it is that the NSA captures EVERYTHING. If something happens later in his life and he is up against it legally, the FBI could conceivably obtain a FISA warrant, search the NSA database and unmask your friend as they go through the process of indicting him.
Tell him to be careful and not to look up the "really, really cool stuff." Here there be dragons!
Final note: NEVER NEVER NEVER go on the dark web. It is a bad and very dangerous place.