How can relativistic effects be the cause of magnetic forces when the normal speeds of carrier charges are so low? I had read here that magnetism arises from a current because of the special relativistic effect associated with the speed of the moving charges in that current.  However that speed is only on the order of 0.024 cm/s, essentially zero compared to the speed of light, so there cannot be a relativistic effect to consider.  Is special relativity the origin of magnetism?
 A: It's because of Lorentz contraction. From the reference frame  of the wire, the charge appears to be moving. From the reference frame of the charge, however, it's the wire that's moving. If the wire is moving at all, then Lorentz contraction will shrink the wire in that reference frame a minuscule amount. This results in an unbalanced number of charges in the wire, producing a magnetic dipole. (charges were balanced $1e^-:1p^+$ but now its something like $1e^-:1.001p^+$, although that's just for illustration. I don't know the exact numbers)
A very good video explanation can be found here.
How electromagnets work
Wikipedia on electromagnets.
A: Special relativity is indeed the origin of magnetism.* As you correctly note, the actual speeds involved in conduction currents in real-life conductors is at crawling pace and it is much smaller than the speed of light. However, most objects tend to be neutral and therefore exert no electrostatic forces on each other, in which case the magnetic contribution, however small, becomes the leading term. In practical examples where magnetic forces are considerable, they are brought about by a large number of charges moving very slowly
This framework is explained in detail, in an intuitive and very readable fashion, by Ed Purcell in chapter 5 of Electricity and Magnetism, which you should go and read. In this post I will not offer a complete proof of the precise statement I endorse (specifically: magnetic forces are the unique way of making electrostatics consistent with special relativity), as that is a complicated calculation to show in detail, well covered by Purcell, and not the spirit of the question as posed. What I will do instead is give some estimates of the forces involved to show that, while relativistic effects at speeds of $\sim0.025\:\mathrm{cm/s}$ $\approx10^{-12}c$ are indeed small, the large values of the electrostatic forces involved can indeed make even such small relativistic effects up to real-world measurable values.

Consider, then, two parallel wires carrying a current of 1 A, separated by 10 cm, so they attract each other with 2 μN of force per meter of wire. This is a perfectly reasonable physical situation, and while the force is not huge it is definitely observable.
Now, suppose that the carrier drift speed in the wires is 0.025 cm/s, as you give. To sustain the fairly high current we're driving in the wires while keeping the carrier drift speed constant, you need a large number of active charge carriers. A carrier density of $n$ carriers per metre, each of charge $e$ and travelling at velocity $v=0.025$ cm/s will carry a current $I=env$, so the charge carrier density is
$$n=\frac{I}{ev}=2.5\times 10^{20}\:\mathrm{cm}^{-1}.$$
This is a pretty reasonable number, since it uses about 1 electron per every 10 sodium atoms (at a reasonable wire density of 0.1 g/cm).
I should note that this is really a sizeable amount of electrical charge involved. The charge density involved in carrying the current is
$$
en=e\:2.5\times 10^{20}\:\mathrm{cm}^{-1}\approx 40\:\mathrm{C}/\mathrm{cm},
$$
and two 40 C charges separated by 10 cm repel (or attract) each other with a force of about 1012 N. (More to the point, the 40 C of charge in 1 cm of one wire feels a force of about 1016 N from an infinite line charge of 40 C/cm at the other wire.) In real-world wires, of course, this electrostatic force is balanced by the presence of the (positive) ionic lattice, which has an exactly equal (and enormous) amount of positive charge which makes the wire electrically neutral.
OK, having said that, we can now bring in the relativity. The mechanism is the very slight length contractions brought about by the relative speeds between the charge carriers on the different wires. Because the carrier drift velocities are very small, the length contraction effects are very small. Howeve, the hugeness of the electrostatic effects I just mentioned can bring even such small relativistic effects into real-world observability.
To be more precise, for a moving carrier in one of the wires, the lattice and the charge carriers on the other wire have different relative velocities: if the currents go in the same direction, the ionic lattice goes the other way, and a similar analysis holds for counter-flowing currents. This difference in velocity causes a slight length contraction, by a factor of
$$
\frac1\gamma=\sqrt{1-v^2/c^2}\approx 1-v^2/2c^2\approx 1-10^{-24},
$$
and a corresponding increase by $\gamma \approx1+10^{-24}$ in charge densities. The complete relativistic analysis is slightly involved and Purcell gives a good version, but the overall dependence as $\sim  v^2/c^2$ in the result is the correct scaling.
Applying this to our wires, the 40 C/cm of current-carrying electrons on our wire now feel a repulsive force of $10^{16}\:\mathrm N$ from the line charge of co-moving electrons, and an almost equal force, of $(1+10^{-24})10^{16}\:\mathrm N$, attracting it to the lab-stationary ionic lattice of the other wire. The electrostatic contributions cancel, and this leaves a force of roughly $10^{-8}\:\mathrm N$ on each centimetre of wire, which is perfectly consistent with the $\sim\mu\mathrm N$ of force per metre of wire that we started with.
So, the takeaway message is: relativistic effects at carrier-charge drift velocities are very, very small indeed. However, normal materials contain huge amounts of balanced charges inside them, and the individual electrostatic forces, which later balance out, are very large. Relativity induces a slight imbalance between them which is proportionately very small, but it is made measurable by the large individual forces involved.

* The word "origin" is a bit contentious. As John Duffield points out (among other points which are incorrect or misleading), the electromagnetic field is a single entity and it is ultimately wrong to do electrodynamics in anything other than a fully covariant, field-tensor-based formulation. However, my opening statement can be made precise: magnetism is the unique way to make electrostatics consistent with special relativity.
A: 
I had read here that magnetism arises from a current because of the special relativistic effect associated with the speed of the moving charges in that current. 

Yes, you can read about that in apparently authoritative sources, but it just isn't true. It doesn't make sense either, because charged particles move rotationally in a magnetic field, and  the length-contraction explanation just doesn't explain that. It's a non-explanation. A fairy tale.    

However that speed is only on the order of 0.024 cm/s, essentially zero compared to the speed of light, so there cannot be a relativistic effect to consider.

Correct. Magnetism arises from relative motion, not from relativistic motion and length contraction. 

Is special relativity the origin of magnetism?

Not in the length-contraction sense. But don't think special relativity is totally divorced from magnetism. See Minkowski’s Space and Time from 1908: 
"In the description of the field caused by the electron itself, then it will appear that the division of the field into electric and magnetic forces is a relative one with respect to the time-axis assumed; the two forces considered together can most vividly be described by a certain analogy to the force-screw in mechanics; the analogy is, however, imperfect".
It's the field, singular, because the electron has an electromagnetic field, and electromagnetic field interactions result in linear electric force and/or rotational magnetic force. Not because of length contraction, but because of the way electromagnetic fields are. See how Minkoski mentioned a screw? Maxwell did too, in On Physical Lines of Force: 
"A motion of translation along an axis cannot produce a rotation about that axis unless it meets with some special mechanism, like that of a screw." 
We don't have the right hand rule for both the current in the wire and screw threads for nothing. Unfortunately this doesn't seem to feature in modern teaching, which rather describes electric force E and magnetic force B as if they're two different fields. This is contrary to the whole thrust of Maxwell's electromagnetic unification. See the electromagnetic field article on Wikipedia and note where it says this: "Over time, it was realized that the electric and magnetic fields are better thought of as two parts of a greater whole - the electromagnetic field..". John Jackson refers to this in section 11.10 of his Classical Electrodynamics textbook: "one should properly speak of the electromagnetic field Fμν rather than E or B separately". However this is in section 11, not section 1.  
