Why is the meter considered a basic SI unit if its definition depends on the second? 
The metre is the length of the path travelled by light in vacuum during a time interval of 1  ⁄ 299792458 of a second. – 17th CGPM (1983, Resolution 1, CR, 97), source

The meter (or metre) is considered one of the seven base units in SI, but since it depends on what is “a second”, shouldn’t it be a derived unit? Why is the meter considered a base SI unit, even though it depends on the definition of a second?
I am aware that other base units such as the ampere also depend on other base units (namely meter, kilogram and second), and this just increases my confusion even more. Perhaps my question boils down to: What property must a unit have so that it is called a base SI unit?
 A: The metre depends not only on the definition of the second, but also on the definition of the speed of light as the quote says.


*

*You could define the second and the metre-per-second (maybe give it a new name in that case so it doesn't "sound" derived) as SI base units and the metre as a derived unit, or

*you could define the metre and the second as base units and the metre-per-second as the derived one.


In any case you'll need two definitions because you want to cover both the dimension of time and that of space.
The choice might be more historical than fundamental (the metre was first - in the original definition the length of a (more or less arbitrarily chosen) specific platinum bar stored in Paris, France just like the kilogram.)
We might all be able to agree that the most fundamental base units would be the metre, second, Coulomb, gram, number-of etc. But as you state yourself, the Coulomb is not chosen to define charge, but rather Ampere defines current, the gram is not defining mass, but rather kilogram is, and the amount of substance is not a simple number-count, but rather defined with the mole.
The SI system is weighed down by convention, tradition and history in this matter. It's point is not really to define things the most fundamental way - rather exhaustively (and of course precisely).
A: OP here.
Instead of choosing an answer to accept, I decided to post my own, collecting what I learned from the great answers by @Wrzlprmft, @Steeven and @MassimoOrtolano (all upvoted!), and organized in such a way that answers my own questions more directly. Thank you all, I learned a lot.

TL;DR - Answer to question in title:
At a first glance, because although it does depend on the second, it is still arithmetically independent from the second (and the other five base units). But looking deeper, all this boils down to historical reasons.

What property must a unit have so that it is called a base SI unit?
First of all, the SI base units were chosen: they are not imposed by nature. Therefore, although quite disappointing, the correct answer to the question above would be "it must be one of meter, kilogram, second, ampere, kelvin, mole or candela". This answer is not fulfilling, though.
They are called "base" units because all units can be derived from them (similarly to the basis of a vector space, for example). Therefore, they are supposed to have these properties:


*

*All units can be written as a combination of the base units and dimensionless constants.

*All base units are arithmetically independent. This means that no base unit can be written as a combination of the other base units and dimensionless constants.
The set "meter, kilogram, second, ampere, kelvin, mole and candela" happen to satify these properties. But many other sets also do.
So this begs the question: Why this specific choice, why not other choices? The answer to this, to me, is quite disappointing as well: it all boils down to measurement convenience and historical reasons. Let's look further into this.
Consider the known "ampere versus coulomb" story. The ampere was chosen as a base SI unit, and coulomb is just a derived unit, obtained as $C = A \cdot s$. Instead, it would be fine to choose the coulomb as the base unit, and the ampere as the derived unit, obtained as $A = C \cdot s^{-1}$. Why the ampere and not the coulomb? Basically, the ampere was chosen for measurement convenience (see linked question to know more). So far so good, since a choice has to be made anyway.
But there is a major caveat in the second bullet above!! It might be surprising at the first, but whether or not two units are arithmetically independent is also historically grown!!


*

*Consider the Magnetic Flux Density (B-field) and the Magnetic Field Intensity (H-field). The first is measured in tesla and the second in ampere per meter. They are arithmetically independent (because tesla cannot be written solely from ampere per meter). We have the relation $B = \mu H$, where $\mu$ is measured in $T \cdot m \cdot A^{-1}$. Well, instead, we could have $T = A \cdot m^{-1}$ and have $\mu$ dimensionless, had science developed in another way.

*Consider the Electric Charge and the Electric Current. The first is measured in coulomb and the second in ampere. They are arithmetically dependent: $C = A \cdot s$. We have the relation $i = \frac{dq}{dt}$, or for the sake of the argument, $i = \alpha \frac{dq}{dt}$ where $\alpha = 1$ is dimensionless. Well, instead, we could have $C$ and $A$ arithmetically independent, and have $\alpha$ be measured in $A \cdot s \cdot C^{-1}$ had science developed in another way.

Why is the meter considered a base SI unit, even though it depends on the definition of a second?
While writing this answer, I realized that this question has actually two interpretations. The unintended interpretation would be: Couldn't other unit be used in its place? The answer is, yes, definitely another unit could have been used, such as the newton. This would be the same story of using the coulomb in place of the ampere, no problem.
Now, to the correct interpretation: Why is the meter considered a base SI unit, even though it depends on the definition of a second? Shouldn't the meter be a derived unit, leaving only the other six units as the base units?
Short answer: It is indeed a base unit. It shouldn't be a derived unit. And this is all due to historical reasons.
First of all, although the meter does depend on the definition of a second, that is not the only thing it depends on. It also depends on this thing called "light" (more precisely, on how fast light moves). Just the fact that it depends on the second is by no means enough evidence that it is a derived unit. We have to look further into it.
Consider the other six base units, from the SI the way it is:
$$\{\text{kg}, \text{s}, \text{A}, \text{K}, \text{mol}, \text{cd}\}$$
Is it possible to write the meter as a combination of the above units and dimensionless constants? Think twice before answering!! You might have said "no!" in your head, but it's not that simple. In fact, this is also a choice. But not a choice that we make easily. Instead, it's a choice that history already made for us. History chose that meter can't be expressed this way. But, quoting @Wrzlprmft:

Had the finite speed of light been a pervasive phenomenon to mankind since the dawn of time, we might have incorporated this strict relationship in our thinking and unit system, always equating a length with the time it takes light to travel that length and never using separate units for length and time.

Therefore, had mankind evolved in a different way, it might be very natural to just write
$$1\text{ m} = \dfrac{1}{299\text{ }792\text{ }458}\text{ s}$$
and have the SI with only six base units.
A: Within the SI system
Let’s first assume the SI system as given. Within this system, the characterising feature of base units is that they are arithmetically independent of each other, i.e., one cannot be obtained from the other using arithmetic operations and real numbers. For example, the hertz arithmetically depends on the second: $1\,\text{Hz} = 1\,\text{s}^{-1}$. By contrast, you cannot write down an equation that has $1\,\text{m}$ on the left-hand side and something that does not explicitly or implicitly contain the unit $\text{m}$ on the right-hand side – the metre is arithmetically independent from the other SI base units.
Choice of the unit system 1: Keeping arithmetics
Given some arithmetic relationship between units, what units you choose as base units is arbitrary, as long as they are arithmetically independent in your system. E.g., if you take the SI system and replace the ampère with the coulomb as a base unit, you still end up with a valid unit system. However, you cannot just add the coulomb to the set of base units and still obtain a valid unit system, as it would then be arithmetically dependent on the ampère and the second. The SI system using the ampère instead of the coulomb as a base unit is due to history.
Choice of the unit system 2: Different arithmetics
What units are arithmetically dependent (and thus the number of base units) is historically grown as well, based on what were well-established experimental facts. A few examples to illustrate this:


*

*Had the finite speed of light been a pervasive phenomenon to mankind since the dawn of time, we might have incorporated this strict relationship in our thinking and unit system, always equating a length with the time it takes light to travel that length and never using separate units for length and time.

*If the relationship between charge, current, and time were not so straightforward, we might as well have defined arithmetically independent units for each of them and later found the empirical relationship:
$$\text{[unit of current]} = k \frac{\text{[unit of charge]}}{\text{[unit of time]}},$$
where $k$ is some constant. Instead we chose our units such that $k=1$, establishing an arithmetic relationship.

*On practically relevant scales, the equality of gravitational and inertial mass is very well established experimentally and a pervasive phenomenon. Therefore we use the same unit to measure both inertial and gravitational mass. Should it turn out that there is a difference between the two, the SI unit system (and any other current unit system) would not be well equipped to describe the pertaining phenomena. If there were a clear difference between the two, we would probably not use the same unit for them.

*In many areas of physics, natural unit systems are used that equate natural constants (such as $c, \hbar, …$) to 1. You can view this as a notational parsimony: If writer and reader know what quantity is measured and what unit system is used, there is no need to write down the quantity. You can also view this as introducing new arithmetical equivalences and thus reducing the number of base units in the unit system.
A: You are confusing two concepts which are actually independent: that of being a base unit and that of the unit definition.
The starting point is that of system of quantities, which is a set of equations (e.g., Newton's law, Maxwell's equations in the rationalized form) specifying the relationships between all the quantities of interest.
Among all the quantities of a system, one can conventionally choose a set of base quantities with the property that each quantity of this set cannot be expressed as a function of the other base quantities, only. That is, I cannot express a length as a function of time, only.
A base unit is then a unit conventionally chosen for a base quantity. The independence between the base units implies that I cannot write 
$$\mathrm{m} = f(\mathrm{s}),$$ 
with $f$ containing only dimensionless constants. However, there can surely be relationships involving the metre, the second and other quantities with dimension, and you can actually use those relationships to define a base unit.
Therefore, the definition of a base unit can depend on other base units, if such a definition is convenient for an accurate realization of the unit, but the definition of a base unit cannot depend on other base units, only.
Remark. Note that the way the unit metre is defined involves a shift of paradigm with respect to the classical way of defining units. This shift of paradigm will become standard in the upcoming revision of the SI (2018).
In the classical paradigm, base units are defined independently from the fundamental constants. If $K$ is a fundamental constant and $[K]$ is its, possibly derived, unit, then the numerical value of $K$ is determined through an experiment as
$$\{K\} = \frac{K}{[K]}.$$
In the forthcoming revision, all base units will be defined in a way similar to that of the metre, so that seven fundamental constant will have an exact numerical value. If $K$ is a fundamental constant, its numerical value $\{K\}$ is set by definition and $[K]$ is indirectly obtained as 
$$[K] = \frac{K}{\{K\}}.$$
A: You are approaching this from a wrong perspective (that is, a scientific one). From  a  scientific  point  of  view,  the  division  of  quantities into base quantities and derived quantities is a matter of an arbitrary convention, and is not essential to physics.
The SI was established recognizing the importance of well-defined units that are easily accessible to all. The intention was to find a set of universally agreed units for measurements in today's society. The units need to be readily available to all, be constant, and easy to measure with high accuracy. 
If the SI considers a unit to be basic or to be derived that does not says much about the nature of this unit or of the magnitude it represent.
A: The meter is considered a basic SI unit because a unit of length is needed, not because of the way it is obtained/derived!
The original meter standard was obtained independent of time. The new definition is being derived with time because we have made great progress in measuring time accurately.  And together with the more convenient access to caesium 133, it allows a much more accurate and convenient way to provide (make) copies (references) of the meter.  
