TLDR: It's not that painting half is optimum, it's that it's sufficient; you don't need more.
We know an amazing amount about 101955 Bennu and its orbit. D.S.Lauretta et al summarize this in their 2015 paper "The OSIRIS-REx target asteroid (101955) Bennu: Constraints on its physical,geological, and dynamical nature from astronomical observations" which includes this about the non-gravitational influence on it's existing orbit:
The Yarkovsky effect was found to be the most significant nongravitational acceleration acting to alter the asteroid’s orbit (Chesley et al. 2014). The Yarkovsky effect is a nongravitational thermal force that results from the way the asteroid rotation affects Bennu’s surface-temperature distribution. The absorption of sunlight, and its anisotropic thermal re-emission, can cause a small thrust (Chesley et al. 2003;Bottke et al. 2006). When thermal forces align with orbital velocity vectors, the Yarkovsky effect produces a steady drift in semimajor axis. Measurement of the Yarkovsky acceleration for Bennu is possible because we have obtained three precise series of radar ranging position measurements over a 12 yr period (ten orbits of Bennu around the Sun). The Yarkovsky effect produces a mean rate of change of Bennu’s semimajor axis of $-1.90(\pm0.01) \times 10^{-3} \rm{AU \: Myr}^{-1}$. Since first being observed in 1999, Bennu has drifted over 160 km as a result of this acceleration
So we have an measurement of the oft-discussed Yarkovsky effect on this asteroid. The effect is an force, hence acceleration, that depends a bit on distance from the Sun, but we can average that over each (slowly changing orbit). Note the displacement due to a constant acceleration grows quadratically with time: remember $d = 1/2 a t^2$?
That means between now and Bennu's close pass in 2135, the Yarkovsky effect moves Bennu by $(117 \rm{yr} / 18 \rm{yr})^2 \times 160\rm{km}$, or about 6,000 km.
The year 2135 pass is interesting because it's close enough to the Earth (0.003 AU) that Bennu's trajectory will be changed significantly. Depending on exactly how it's changed, that can bring Bennu into contact with Earth later on. Steven Chesley et al's 2014 paper "Orbit and Bulk Density of the OSIRIS-REx Target Asteroid (101955) Bennu" looks at this in great detail, identifying a large number of small "keyholes" that, if hit, will cause a later collision:
There are a lot of them, but they're all very very small compared to the 6,000km deflection due to the Yarkovsky effect. Note that zero isn't on the axis: Bennu isn't now aimed at any of them. The 2182 window is the closest, and the next one is 50,000km (note scale of figure) away.
If we had sufficiently accurate information, and determined that Bennu was headed for one of the keyholes, even a 20% change in the Yarkovsky effect would allow us to prevent a (much later) collision with Earth.
Now let's return to the Gizmondo article, which quotes NASA's Michael Moreau as saying (emphasis added)
“Even just painting the surface a different color on one half would change the thermal properties and change its orbit”
I think Moreau is aware of the size of the Yarkovsky effect, and realizes that only a (comparatively) small modification of it would be sufficient. So the "only .. half" is an expression of "we don't have to cover the whole thing 100% using a fine brush, we just have to modify the surface somewhat"
So it's not that "painting half of Bennu be more effective than painting all", it's that it's not necessary to do any more than painting half to move away from the (future) collision.
More detail on the orbital dynamics: The Yarkovsky effect in a specific case depends on the rotation of the asteroid. For a retrograde rotating (yes, we really know a lot about Bennu) asteroid, you can think of it as propulsive along the tangent to the orbit but against the motion. That's slowly lowering the orbit, lowering the semi-major axis by about $284 \pm 1.5 \rm{m/yr}$ (Chesley; note the sign should be negative in several places): If the Yarkovsky effect is increased, Bennu will be lower when it intersects Earth, closer to the Sun, though not necessarily closer to Earth. In the process, it's also changing the orbital period: this can add up over many orbits (many years) to make a much larger difference in position, which corresponds to a difference in arrive time at a point.
When working out whether a particular event, e.g. a keyhole passing, happens you have to consider both position and passage time. Chesley discusses this on page 17:
Table 5 details the effect of various differing models on the b-plane coordinates $(\xi_{2135},\zeta_{2135})$ of the close approach at the last reliably predicted Earth encounter for Bennu, which takes place in 2135. The b-plane is oriented normal to the inbound hyperbolic approach asymptote and is frequently used in encounter analysis. The (ξ,ζ) coordinates on the b-plane are oriented such that the projected heliocentric velocity of the planet is coincident with the −ζ-axis. In this frame the ζ coordinate indicates how much the asteroid is early (ζ < 0) or late (ζ > 0) for the minimum possible distance encounter. In absolute value, the ξ coordinate reveals the so-called Minimum Orbital Intersection Distance (MOID), which is the minimum possible encounter distance that the asteroid can attain assuming only changes to the timing of the asteroid encounter. For a more extensive discussion of these coordinates see Valsecchi et al. (2003) and references therein.
(Table continues in original paper with lots of rows for other really interesting effects, too) Comparing the $\Delta \xi$ value of a few tens of km and the $\Delta \zeta$ value of thousands of km shows that the largest effect is the delay: The Yarkovsky effects makes Bennu arrive late.
And that's why the horizontal axis of Fig 5 above is in terms of $\zeta_{2135}$; that's the delay that would change if you've changed the size of Bennu's Yarkovsky effect.
More detail on painting: White or black?
Either "white" or "black" paint would modify the size of the Yarkovsky effect on Bennu's orbit. From Lauretta et al:
The geometric visible albedo (pV) of Bennu is wellconstrained. Using the relationship 2.5 log pV =15.62 - 5 log D - H, where H is the absolute magnitude (from Hergenrother et al. 2013) and D the asteroid size (from Nolan et al. 2013), constrains pV to 4.5 +/- 0.5%. Applying a known correlation between the slope of the linear phase function and the albedo of asteroids (Belskaya and Shevchenko 2000; Oszkiewiczet al. 2011), yields an albedo of 3.0–4.5% based on Bennu’s phase function slope of 0.040 mag/deg (Hergenrother et al. 2013). Near-infrared spectroscopic data show a thermal tail longward of 2 microns (Fig. 5),consistent with an albedo of 4 +/1 1% (Clark et al. 2011). Spitzer photometric measurements combined with visible photometry constrain the albedo to 4.3 +/- 0.3% (Emery et al. 2014). OSIRIS-REx has adopted a geometric albedo of 4.5 +/- 0.5% for Bennu based on all of these independent determinations.
It's easier to raise a 4% albedo (with white paint) than to lower it (with black paint). But there's more to the question than that, because different paints require different amounts of mass to achieve coverage, have different procedures for application, etc. Both lighter and darker pigments have been studied: for example, S Ge and Hyland studied both carbon black (dark) and TiO2 (white). (See also Shen Ge thesis) Dark pigments tend to be more effective per kg of paint, but I don't think there's a complete understanding of all the alternatives. Research is continuing.
Which is better for avoiding a collision, more or less Yarkovsky effects?
It's important to note that right now, the answer is "neither". As far as we know, Bennu is not colliding with Earth in the next couple centuries. But we might be wrong: Extrapolating motion for that long is hard (take another look at the list of effects in Figure 5 of Lauretta et al). We could have it wrong. So at some time in the future, we might discover that Bennu is headed for Earth.
Most likely, that (eventual) collision will follow on an earlier Earth interaction that diverted Bennu to the collision. That's one of the keyholes mentioned above. In some sense, that makes the diversion problem simpler, but the extrapolation problem harder.
The diversion problem is simpler because a keyhole is much smaller than Earth. Typically a few km to maybe a couple hundred km wide, it takes much less of a change in orbit to miss one that to miss Earth. (You have a bit less time, due to the keyhole interaction being earlier, but generally the ratio of sizes is more important).
But note that if you modify the asteroid, you modify the path before and after the keyhole encounter: Worst case, you can miss the keyhole, but the asteroid will take a different path that now results in a different collision, and you've got to do the whole thing again with either more or less (!!) time to pull it off. Being able to predict those new paths accurately is very important.
So the mission planning is hard. We're getting better at predicting orbits (and asteroid perturbations), which is good. We have a bit of time AFAIK. And when we find that we really do need to modify an orbit to avoid a collision, that modification might involve either more or less Yarkovsky acceleration, so we might be sending out for either white or black paint.
