The following anime share the use of the semicolon as part of their names:
What is the reason for using the semicolon within the titles of these anime?
The best answer that I could make out myself for the use of the semi-colon in the names:
The semicolon in the names are used as a form of "pertaining to," however "not in possession of." Meaning--for example--"Steins" is related to "Gate" however, "Steins" is not in possession of "Gate." (This applies to the other two names as well.) We can see this because "Steins" does not make use of an apostrophe, which denotes possession.
We can see this by also rewording the titles of the anime:
Gate of Steins
Head of Chaos
Notes of Robotics
In this case, we see that "Steins," "Chaos," and "Robotics" all are related to the the preceding words in the reworded titles, just the same as if we made use of the semi-colon instead of, "of," in the original titles.
Oh no, I'm two years late to the discussion. While browsing on Steam, I came across STEINS;GATE and wondered why the semicolon was used in the title. I started reading reviews of the game but didn't find anything and so I Googled it and came to this page. From the reviews, STEINS;GATE sounds like an excellent visual novel, and so my comments here shouldn't reflect on the creators of the game; they are quite talented.
It would be interesting if the use of the semicolon here is an example of how World English is evolving. If all through the continent of Asia, for example, the semicolon in English has come to mean "related to but not possessed by" then I'd understand its use here. If not, however, then this is just a typographical error. I've had English professors whose hair would stand on end if they saw the semicolon used this way. If a student submitted a paper with a proper noun styled like this, those professors would circle the semicolon vigorously in red and make a brief comment with a lot of exclamation marks. That would be an overreaction, of course, but you can see that the construction is so odd that even I'm motivated to comment on it, two years late even.
FatalSleep provided a very clearly written answer, but in contemporary Standard English, the semicolon isn't used to connect nouns unless they are in a complex series. If you have a series of three or more elements, and one or more of those elements have three or more elements of their own, 1) a semicolon is used to separate the main elements in the series, to avoid confusion. The only other time a semicolon can be used correctly in Standard English is 2) to connect two independent clauses with no conjunction when they aren't serial clauses. The last sentence of my first paragraph is an example of that. These are the only two correct uses of the semicolon. The semicolon isn't a soft colon, even though that's what the name looks like; it's a hard comma, with only two uses.
In contemporary usage, STEINS:GATE is more familiar and would've been a better choice. The irony is that, because STEINS modifies the noun GATE, nothing is needed to show that they are connected. STEINS GATE is referring to only one gate, closely associated with STEINS but not possessed by them because there's no apostrophe. A variation of that would be SteinsGate, using camel caps, to indicate the same thing. Even STEINS-GATE would've been better than STEINS;GATE. But they've also created CHAOS;HEAD and ROBOTICS;NOTES. My professors would've burst a blood vessel over those. What were the game creators thinking?
If lots of people start using a semicolon to connect words because they think it looks cool, then eventually it will become standard usage in World English and no one will complain. Eventually. Users of Standard English will wonder why, but they'll just shrug and say "Like, whatever."