It depends on the system, since the creators' attitudes vary so much. Most companies are not eager to give out these licenses as that makes you competition for their products, but there are enough exceptions to make licensing a system a reasonable route to go. As a rule, you will need some kind of written agreement to license a system.
For GURPS specifically, SJ Games does allow third-party products that are "Powered by GURPS". There is little information on that page about what sort of licensing terms SJG will ask for, but they have been and appear to continue to be willing to work out private licensing deals if people are interested. If your friend really wants to consider GURPS, they can certainly contact Steve Jackson Games and find out what licensing the system entails so they can decide whether it would work for them.
Assuming your friend owns the other creative properties, you could also ask Steve Jackson Games if they're interested in creating the game for you. It's not clear how much creative control would be retained in that case, but if your friend is going to be contact SGJ anyway (and is interested in someone else doing the job) it may be worth finding out at the same time if that's a possible route to publication.
Apart from GURPS, the most hassle-free option is using a system that is licensed under the Open Game License (OGL), one of the Creative Commons licenses (CC) compatible with your friend's aims, the Gnu Free Document License (GFDL), or another open license. The advantage of such licenses over a negotiated license is that the license can be used immediately just by following the terms it lays out, without having to negotiate directly with the rights-holder of the system. There is a list of open games at Wikipedia. The most notable of these are:
System Reference Document (d20), of which there are hundreds of games and supplements, Dungeons & Dragons foremost among them.
FATE, including Spirit of the Century (pulp), Diaspora (sci-fi), Starblazer Adventures (pulp sci-fi), Dresden Files RPG (modern wizardry), and Legends of Anglerre (fantasy). Some of those I know have their own open licenses, so there's the option of extending an existing game as well as building an independent FATE game.
Fudge, which is a tool-kit system that is highly hackable, and of which FATE is a descendant.
A variety of old-school D&D clones, including Labyrinth Lord, Swords & Wizardry, OSRIC, Dark Dungeons, and more. These are notable as examples of how to bend, fold, spindle, and mutilate the d20 System Reference Document into a game that only superficially resembles D&D 3rd edition. A gonzo sci-fi example is Mutant Future, which is an adaptation of the Labyrinth Lord system.
Another option is to license one of the systems owned by publishers that advertise very liberal licensing attitudes. Two in particular have stood out to me:
Savage Worlds is particularly known for being licensable under an "ask us, and we'll probably say yes" attitude, which is responsible for the ridiculous number of high-quality settings published for Savage Worlds by third parties in the last couple of years.
HeroQuest 2nd Edition can be licensed under similar terms and without much effort. Being a genre-agnostic system for simulating plots (rather than world physics), it's no work to port to a setting but its story-logic design isn't to everyone's taste.
There are more options of course, as these are just the ones that stick out for me.
And of course, you could always write a game from scratch! There is a long and honoured tradition of learning from the designs of existing games and incorporating that into a new game, so licensing isn't the only option if your friend has any desire to do system design. The actual writing could be done in a week or two if you put your mind to it. The playtesting would take a while anyway, no matter what game system you chose to adapt.
While I'm not a lawyer, I'd suspect that part of your liability here is determined by the method you do the 5e updates. Are you talking about republishing them, including maps and flavor text? If so, I'd be very cautious, especially given WotC's proclivity to shut down 3rd party resources for older versions of the game.
However, a simple listing of changes gives you a much larger footing here. Many of these changes should revolve around monster and trap changes, so I'm thinking something like
p. 2, Kobolds become 5e Kobolds (MM p.whatever)
p. 3, Greater Golem becomes Iron Golem (MM p.whatever)
and etc.
While Wizards might still send you a cease and desist, it seems an awful lot less likely. The maps and original text are almost certainly under copyright, so I'd be extremely cautious about republishing them.
Best Answer
In terms of the GURPS part, assuming this is not for sale and just posted publicly on the Web, your use would be dictated by the Steve Jackson Games Online Policy. This allows you to make adventures and stuff but not things that require a restatement of the GURPS rules - so you'd want to be careful that your "fan book" doesn't do that.
In addition, the online policy warns you about the second problem -
Making a fan book for those video game properties is technically not legal (though they may have a fan use statement of their own that lets you). However, many netbooks exist out there in the wide world because of a mix of flying under the radar/no one caring.
Super Legal Answer:
I am not a lawyer, hire one to advise you! The most "correct" answer from the point of view of a cover-your-ass legalistic society. They'll just tell you not to do it, though. "Fun versus a minute amount of legal risk? What is this 'fun'? Don't do it or you may end up in PMITA federal prison. That'll be $500." Save the money and just don't do it if you're risk averse.
Semi Legal Answer:
Contact both SJG and the game IP owners asking for permission to do this. Of course, it'll be hard to actually contact anyone about those old games, and the routine answer is "no" for more of those CYA legal reasons above. Sadly most media companies haven't figured out yet that stuff like this just helps to increase the buzz around and value of their brands.
The Most Common Answer:
Do it, but be prepared to take it down if any of the parties involved demand you do. Some may. Carries marginal risk of a vengeful company trying to ruin your life even if you take it down, which best as I know has never happened over a free fan book.