I can think of several possibilities that might become feasible in the next century.
Dropping a nuclear reactor into Earth's core. A large, very hot nuclear reactor could conceivably melt its way through the crust and mantle all the way down to the core. The reactor could be designed to target a particular destination in the outer core, which would be chosen based on suitable measurements and modeling. Upon arriving, it would increase its power output dramatically. The heat would disrupt the dynamo that drives Earth's magnetic field, causing a geomagnetic reversal.
How much nuclear fuel would be required? As an order-of-magnitude estimate, we might guess that the reactor would need to produce about 50 TW (the approximate total heat flow from Earth's interior to the surface) for a decade to have a significant effect. To do this, it would need to burn about
$$\frac{(50 \times 10^{12} \text{ W}) \cdot (10 \text{ years}) \cdot (235 \text{ g/mol})}{(\text{Avogadro's number atoms/mol}) \cdot (200 \text{ MeV/atom})} \approx \text{200,000 tons}$$
of fuel. The world currently produces around 60,000 tons of natural uranium per year. Although only about 0.7% of that uranium is the fissile isotope uranium-235, a well-designed fast breeder reactor could convert (most of) the remaining uranium-238 into fissile plutonium-239 and then burn it. So this seems difficult but potentially feasible, especially if we assume some technological advances over the next century.
Building a very large superconducting magnet. A loop of superconducting wire encircling the entire planet could generate a magnetic field strong enough to directly counteract the natural field from the core.
How much superconducting wire would we need? Suppose we need to generate a field of 1 gauss at the center of the current loop. That's about twice the strength of the planet's natural magnetic field, so it should be enough to reverse the field's direction. The required current would be about
$$\frac{\text{1 gauss} \cdot 2 \cdot \text{radius of Earth}}{\mu_0} \approx 10^9 \text{ A}$$
Suppose that we are building the magnet from magnesium diboride, a reasonably high-performance superconductor that can be made from commonly available elements. If we assume a current density of $500 \text{ kA/cm}^2$, which is the reported critical current density for a sample of high-quality bulk magnesium diboride, the wire would need to have a cross-sectional area of $0.2 \text{ m}^2$. Since the density of magnesium diboride is about $2.6 \text{ g/cm}^3$, the total mass of superconductor required would be around
$$0.2 \text{ m}^2 \cdot (2 \cdot \pi \cdot \text{radius of Earth}) \cdot 2.6 \text{ g/cm}^3 \approx \text{20 million tons}$$
Every year, the world currently produces around a million tons of magnesium and a few million tons of borates. Converting all of this raw material into superconducting magnesium diboride and building a suitable refrigeration system would be difficult and very expensive. But if we really wanted to do it, a century would be more than enough time.
Generating a large electrical current in the magnetosphere. Instead of building a new superconducting magnet, we could take advantage of the large body of conductive plasma that already exists around Earth: the magnetosphere. Again, a sufficiently large electrical current (around $10^9$ amps) could generate a magnetic field strong enough to counterbalance the natural field and trigger an effective geomagnetic reversal.
There are a number of ways to create electric currents in a plasma. For example, one can use an antenna array to broadcast radio waves parallel to the existing magnetic field. The waves are absorbed through Landau damping, exerting a force that drives an electric current. As the generated current becomes large, the direction of the overall magnetic field rotates. One can then move the antenna array to transmit parallel to the new field direction, and continue the process until the field has completely reversed.
Could we create a strong enough current in this way? It's difficult to accurately estimate the power required, as many processes in the magnetospheric plasma are nonlinear. We'd certainly need something bigger than HAARP, but perhaps not that much bigger. The solar wind (tens of gigawatts) is enough to drive the ring current of about 10 mega-amps, despite very low efficiency. So a well-designed space-based antenna array, in a high orbit (for lower plasma resistivity), with a few gigawatts of total transmit power, might possibly be sufficient.