Which is (more) correct?
- (I am with the) Laboratory of Biochemistry of Hydrobionts.
- (I am with the) Laboratory of the Biochemistry of Hydrobionts.
Is there any rule for using or not using the before Biochemistry?
When you say "The laboratory", you are referring to a specific object, which fulfils the role expressed by laboratory. Contrast this with "a laboratory", which can refer to any object that fulfils the role conveyed by the word laboratory.
Biochemistry refers to a general topic or idea—not to a specific object, nor to any object.
is correct usage.
On the other hand, since this is also the name of a thing,
would be correct as well.
"The" is not needed in this case. It doesn't change the meaning of the sentence. What would change the meaning of the sentence would be a "the" infront of "Hydrobionts" as in
"The" specifies which "hydrobionts" is being referred to. The Hydrobionts, or just Hydrobionts, etc.
Actually @Alenanno provided a nearly complete answer in the question comment.
The first rule of precedence is to use the official title of the institution or facility. Thankfully, that simplifies concern related to grammatical correctness!
One needs to be careful not to imply an official name for something that does not have one, thus incorrectly conveying a greater sense of prestige or status by using the definite article.
For an actual place name, I would expect something such as "The Hydrobionts Biochemistry Laboratory". It sounds more natural with a less archaic usage to me e.g. "The Computational Linguistics Research Laboratory". Regardless, always use the official title whatever it may be, and with the first letter of the first word capitalized:
If the facility name is "Laboratory of Biochemistry of Hydrobionts" or "Laboratory of Biochemistry of the Hydrobionts", then do NOT use the definite article. In that situation, you would say
because that is the name of the institution.
If there is no proper name, but you work for the laboratory that is affiliated with the institution or department of "Biochemistry of Hydrobionts", then do not capitalize "the". You would say
It's going to depend on the context. Is it the Laboratory of Biochemistry which is studying Hydrobionts, or a "general" (i.e. not specialized in biochemistry) laboratory which is studying the biochemistry of hydrobionts?
If it's a biochem lab, drop the the before biochemistry. If it's a general lab, keep the definite article to show it's the focus of the research — the biochemistry of the hydrobionts.