Subjective principle in the Groundwork
Let me quote the footnote that actually defines what subjective principle actually means for Kant:
A maxim is the subjective principle for action, and must be distin-
guished from the objective principle, namely the practical law. The former
contains the practical rule that reason determines in accord with the conditions of the subject (often its ignorance or also its inclinations), and is thus
the principle in accordance with which the subject acts; but the law is the
objective principle, valid for every rational being, and the principle in
accordance with which it ought to act, i.e., an imperative. (Groundwork for the Metaphysics of Morals, Ak. 4:420-1 fn., Cambridge p. 37 fn., bolded by me)
The objective principle (after which the subject ought to act by virtue of being rational) may therefore become the subjective principle of a given subject simply by the subject itself, in choosing it as its principle of acting.
It is objective as it is a rational insight that it has to be a (read: general/universal) principle of the will (of all finite rational beings), no matter if the subjective (read: particular) will chooses it or not.
Subjective principle in the third critique
Now this transcendental concept of a purposiveness of nature is neither a concept of
nature nor a concept of freedom, since it attributes nothing at all to
the object (of nature), but rather only represents the unique way in
which we must proceed in reflection on the objects of nature with the
aim of a thoroughly interconnected experience, consequently it is a
subjective principle (maxim) of the power of judgment;... (Critique of the Power of Judgement, 5:184, Cambridge p. 71)
Here, Kant speaks of 'concepts of nature' and 'concepts of freedom', which 'attribute to the object' of nature/freedom, i.e. are constitutional for these objects. Subjective principles in contrast express how 'we proceed' in order to reach a given (and may it even be necessary, as eudaimonia and 'thoroughly interconnected experience') aim. They are therefore merely regulative.
Another possible (but possibly confusing) way to put it would be the following:
Objective principles are constitutional for realising objects of knowledge, where the representation [Vorstellung] is determined by the object. (Object -> Objective principle -> Representation)
Subjective principles are (in some sense) 'constitutional' for realising objects of the (particular, subjective) will, where a given representation [Vorstellung], and may it be a (kantian!) idea, regulatively determines the object that is produced by its realisation. (Representation -> Subjective principle -> Object)
That should (hopefully!) clarify why the objective principle in fact has to be a subjective one when it comes to practical concerns, because there always the representation (object of the will) precedes the object (outcome of the act in nature) within a particular subject. Particular willing is always the willing of a particular subject.
It is subjective in the sense that it is judged by a particular subject as the principle of this particular subject. This choice does not contain any implications for other subjects, e.g. rational beings as insights. But insights into the nature of subjectivity (of rational beings) very well have implications for particular subjects, independent of their particular 'conditions', and are in this sense objective.