I think (and I broadly agree here with Eckart Förster, whose book The Twenty-Five Years of Philosophy will be quoted here for reference) that Hegel roughly follows Kant's understanding of science. Therefore, clarifying what Kant wrote about science may help to elucidate how this intertwines with the idea of a single principle.
Kant himself on science
Kant thought that science is, as opposed to a modern understanding, a set of constructible, certain sentences. Certain, however, they can only be if the way they are obtained itself is certain, which for him means by and from principles that are themselves certain. Förster draws on these two main aspects:
First, according to Kant science is not merely a rhapsodic collection of propositions, but a totality of knowledge ordered according to principles and hence systematic. Systematicity is only a necessary condition for science, however, not a sufficient condition. For if those principles are merely empirical in nature, what we have is a systematic doctrine, but not a science in the proper sense. Secondly, therefore, the principles must be associated with a “consciousness of their necessity” (4:468). Hence they cannot be principles discovered on the basis of induction, mere generalizations from experience; they must be capable of being known a priori. “All proper natural science therefore requires a pure part, on which the apodictic certainty that reason seeks therein can be based” (4:469). (Förster, 2012:67, bolded mine)
Science proper here only consists in the pure part. That is why in his introduction to the Anthropology, he dismisses psychology as not being a science proper in any way, missing this pure part.
Kant on philosophy as science
It is the main point of the whole critical endevour of Kant to bring metaphysics (or philosophy as such) back on the "secure course of a science":
Metaphysics - a wholly isolated speculative cognition of reason that
elevates itself entirely above all instruction from experience, and that
through mere concepts (not, like mathematics, through the application
of concepts to intuition), where reason thus is supposed to be its own
pupil - has up to now not been so favored by fate as to have been able
to enter upon the secure course of a science, even though it is older than
all other sciences, and would remain even if all the others were swallowed up by an all-consuming barbarism. (Kant, CPR, B xiv)
Now the concern of this critique of pure speculative reason consists
in that attempt to transform the accepted procedure of metaphysics, undertaking an entire revolution according to the example of the geometers and natural scientists. It is a treatise on the method, not a system of
the science itself; but it catalogs the entire outline of the science of
metaphysics, both in respect of its boundaries and in respect of its entire internal structure. (Kant, CPR, B xxii)
From Kant to Hegel
Förster has some four chapters on that, and I cannot refer to all the content discussed. The main core points of Kant prevailed, and it had been a major criticism that he presented several principles (for each faculty of knowledge one), trying to show how these can stand side by side without contradiction.
Most critics argued that there has to be a single, unifying principle. Basically, the idea why a systematic Wissenschaft has to be derived from a single principle is easy: Say we have two a priori certain principles, to take the easiest case. Now, if we want to apply them, they are either materially identical (single principle), or we will have to decide which principle is to be applied. For deciding that, we would need to have a third principle, which would need a justification of its own. The main idea of this dilemma can be found in my answer here as well.
This lead to various attempts of finding a single principle. Förster writes on Fichte:
For Fichte characterized his philosophy, which he now referred to as Wissenschaftslehre, as the discipline which philosophically grounds the possibility of every other science. Since every science must have a systematic
form, and since such form can only be derived from a first principle, the
Wissenschaftslehre must at the same time establish principles for every other science. Those principles must, if they are truly to be principles, be incapable of further proof: “All those propositions which serve as first principles of the various particular sciences are, at the same time, propositions indigenous to the Wissenschaftslehre. Thus one and the same proposition has to be considered from two points of view” (GA I,2:128; W 1:56) (Förster, 2012:175).
This "one and the same proposition" was A=A, the proposition of identity. This has led to criticism by e.g. Hölderlin, who was highly influential on Hegel, his former fellow student in Tübingen and very good friend (they met 1797 in Heidelberg and discussed these things a lot).
I will quote secondary literature here, as this is easier than carving it out of Hegel's text and give interpretation and context. Hegel saw philosophy as a science:
For Hegel, it is the “scientific” in the
sense of the theoretical whole [he links Hegel's understanding of science with 'totality' just before] that determines not the empirical but the
ontological truth of a sentence. He knew very well the difference between
physics and philosophy, for example, although he often opened himself to
ridicule by attempting to express the philosophical significance of physics.
But this was precisely his intention: to provide a philosophical explanation
of philosophy as an expression of the totality of the human spirit. Hegel
accepts the scientific nature of philosophy, but he distinguishes philosophical
from natural science. (Stanley Rosen, *The Idea of Hegel's "Science of Logic", pp. 5-6)
Rosen is quite explicit about that Hegel is looking for something like a fist principle:
Like Hegel, Fichte is obsessed with the need to arrive
at a presuppositionless beginning, in order to transform philosophy into a
deductive or systematic science. He begins his search for the “first, strictly
unconditioned principle” with a distinction between thinking as activity and the principle as an expression of that activity (Stanley Rosen, *The Idea of Hegel's "Science of Logic", p.196)
Regarding the nature of this single principle, this case is clear to some extent if you know Hegel, but hard to track down in a single quote: It is (Absolute) Geist, the all-encompassing conscience. The closest I could find to verify that is the following, effectively describing Geist as ultimate condition of the unity of object and subject, which can only be grasped by looking at the whole of history:
Hegel’s goal to bring together understanding and reason in such a way as
to show that the ideals of the latter are achieved or fully manifested in the
structure of the former. This in turn depends upon the transformation of
the synthetic unity of apperception from a logical condition of consciousness
to a principle of subjectivity, namely, the principle that underlies the
unity of subject and object (Stanley Rosen, *The Idea of Hegel's "Science of Logic", p.22, emphasis mine)
The understanding of what it means to be Wissenschaft, i.e. something that through its methods literally translated creates knowledge [Wissen = knowledge; schaffen = to create], had been quite different. Hume's skepticism led to the understanding that all phenomenal, empirical data is not reliable enough to constitute knowledge in this very peculiar sense.
And what Hegel thought to have accomplished is to present a method that, through its application, is able to get rid of empirical illusions, leading to an understanding of how reality actually is, i.e. to knowledge. Therefore, in this sense, it is Wissenschaft. And it is Wissenschaft der Logik as the knowledge is gained through a very specific methodology he calls logic, although this may differ a lot from contemporary understandings.
Long story short: If terms in historical philosophy do not make any sense, try to look up their historical and, in philosophy, often times technical meaning.