What are the strings in string theory made of?
OP wrote(v4):
[...] Strings in string theory also seem to possess a rather complicated and certainly non-trivial suite materials-like properties such as length, rigidity, tension, and I'm sure others (e.g. some analog of angular momentum?). [...]
Well, the relativistic string should not be confused with the non-relativistic material string, compare e.g. chapter 6 and 4 in Ref. 1, respectively. In contrast, the relativistic string is e.g. required to be world-sheet reparametrization-invariant, i.e. the world-sheet coordinates are no longer physical/material labels of the string, but merely unphysical gauge degree of freedom.
Moreover, in principle, all dimensionless continuous constants in string theory may be calculated from any stabilized string vacuum, see e.g. this Phys.SE answer by Lubos Motl.
OP wrote(v1):
What are strings made of?
One answer is that it is only meaningful to answer this question if the answer has physical consequences. Popularly speaking, string theory is supposed to be the innermost Russian doll of modern physics, and there are no more dolls inside that we can explain it in terms. However, we may be able to find equivalent formulations.
For instance, Thorn has proposed in Ref. 2 that strings are made of point-like objects that he calls string bits. More precisely, he has shown that this string bit formulation is mathematically equivalent to the light-cone formulation of string theory; first in the bosonic string and later in the superstring. The corresponding formulas are indeed quadratic a la harmonic oscillators (cf. a comment by anna v) with the twist that the "Newtonian mass" of the string bit oscillators are given by light-cone $P^+$ momentum. Thorn was inspired by fishnet Feynman diagrams (think triangularized world-sheets), which were discussed in Refs. 3 and 4. However, string bit formulation does not really answer the question What are strings made of?; it merely adds a dual description.
References:
B. Zwiebach, A first course in String Theory.
C.B. Thorn, Reformulating String Theory with the 1/N Expansion, in Sakharov Memorial Lectures in Physics, Ed. L. V. Keldysh and V. Ya. Fainberg, Nova Science Publishers Inc., Commack, New York, 1992; arXiv:hep-th/9405069.
H.B. Nielsen and P. Olesen, Phys. Lett. 32B (1970) 203.
B. Sakita and M.A. Virasoro, Phys. Rev. Lett. 24 (1970) 1146.
The question "what is xxx made of" is really asking "what can xxx be decomposed into"?
For example we know matter is made of atoms because it can be decomposed into atoms. We know atoms are made of electrons, protons and neutrons because atoms can be broken down into electrons, protons and neutrons. But electrons can't be decomposed into anything, so it's meaningless to ask what an electron is made of. We can ask what an electron's mass is, or its energy, or its spin, etc etc, but to ask what it's made from is a question that has no answer.
Exactly the same applies to a string. It is an object that has properties, but it's meaningless to ask what it's made from because it can't be broken down into anything.
Lenny Susskind explains that the answer to this question depends on the parameters of the theory at 1:10:50 to the end of this video.
He makes use of the fact that the question if strings are fundamental or if they are composed of something else is analogous to the question if in electrodynamics, electrons or magnetic monopols have to be considered fundamental to be able to develop a perturbation theory with Feynman diagrams. It can be shown that magnetic charges $q$ and electric charges $e$ are related by
$$ e\, q = 2 \pi$$
This means, that if the charge of the electron (and therefore their mass) is small, the charge of the magnetic monopoles (and their mass) is huge and vice versa. If the charge and the mass of the electron are small, the electron is considered fundamental and a converging theory (QED) can be developed because the coupling constant $e$ is small. At the same time the magnetic monopoles are heavy complicated things composed of whole bunches of photons and magnetic charges because the coupling constant $q$ is large. This regime corresponds to what we observe with QED being a weakly coupled theory and the magnetic monopoles (if they exist) being to large to be observed. Increasing the electric charge of the electron would lead to a transition to a situation, where the electons become heavy and complicated and in this case it would be more useful to consider a quantized electromagnetic theory with the ligth magnetic monopoles described as fundamental particles.
A similar relationship as described to hold for the pair of electric and magnetic charges exists in string theory between fundamental (f-) strings and D-branes. Depending on the parameters of the theory, either it is more appropriate to consider the D-branes as complicated heavy things composed of fundamental strings, or the D-branes are light and fundamental whereas strings are heavy and complicated things composed of D-branes. The technical term describing this ambiguity is S-duality.
In summary, a unique and universally valid answer to the question what strings are made of can not be given; it depends on the parameters of the theory and the context if it is more useful to consider strings or D-branes as fundamental.