It looks like you're new here. If you want to get involved, click one of these buttons!
The general approach that we seem to agree on for better predictions of ENSO is to try to find patterns or correlations amongst various data sets.
This discussion topic is meant to collect information and analyses for the known correlation between the Quasi-Biennial Oscillation (QBO) of stratospheric wind speeds and ENSO, see [1,2,3,4,5] refs at the bottom of this post.
What I find intriguing about this correlation is that the QBO periodicity is much stronger than the erratic ENSO periodicity. One would think this would have implications for predictability of ENSO -- since a stronger period is more predictable than a weaker, and since QBO is thought to drive ENSO in some way, we may be able to isolate a component of the time series. And if this forcing is strongly periodic, it may be used to project into the future.
The reason QBO is thought to be a driver is that the QBO winds downwell over the Pacific Ocean as they cycle. One can see this in the speed vs altitude plots, where a higher atmospheric pressure corresponds to a lower altitude.
This tends to push on the Pacific Ocean surface with the same cycle, causing water to pile up in the windward direction periodically.
The QBO has an average period of about 28 months since data collection started in 1952 (data link), which explains the quasi-biennial aspect. However the cycles show a measurable amount of jitter, which is a fluctuation in a given cycle's period. One question to consider is whether this jitter is random or shows an extra periodicity, which may be due to tidal beating (?) see http://contextearth.com/2014/06/17/the-qbom/ for some evidence that I collected.
The recent claim in  is that the ENSO and QBO time series have almost aligned over a recent 5-year interval. See the figure below.
That near-correlation could be just happenstance as in general the waveforms don't align, with the ENSO being much more erratic and the QBO showing a stricter periodicity. The other characteristic of QBO is that the peaks are generally broader and more flat-topped than the valleys. But now that I look at what others have plotted, the asymmetry is not quite as apparent (from here)
Yet, this asymmetry is not generally seen in ENSO.
I have been analyzing using the fundamental frequency of QBO as a driver to a nonlinear DiffEq equation model of ENSO and the results are intriguing enough that I have pursued this over several ENSO data sets such as SOI and ENSO proxies. Start here and follow the links backwards. The general finding is that periods close to the QBO period appear to be strong candidates for a forcing function, but the non-linear function causes a transformation that the response is much more erratic.
That's the general idea and I will post more analyses related to QBO below this entry.
 N. Calvo, M. A. Giorgetta, R. Garcia‐Herrera, and E. Manzini, “Nonlinearity of the combined warm ENSO and QBO effects on the Northern Hemisphere polar vortex in MAECHAM5 simulations,” Journal of Geophysical Research: Atmospheres (1984–2012), vol. 114, no. D13, 2009.
 M. Geller and W. Yuan, “QBO-ENSO Connections and Influence on the Tropical Cold Point Tropopause,” presented at the AGU Fall Meeting Abstracts, 2011, vol. 1, p. 03.
 S. Liess and M. A. Geller, “On the relationship between QBO and distribution of tropical deep convection,” Journal of Geophysical Research: Atmospheres (1984–2012), vol. 117, no. D3, 2012.
 W. M. Gray, J. D. Sheaffer, and J. A. Knaff, “Inﬂuence of the stratospheric QBO on ENSO variability,” J. Meteor: Soc. Japan, vol. 70, pp. 975–995, 1992.
 J. L. Neu, T. Flury, G. L. Manney, M. L. Santee, N. J. Livesey, and J. Worden, “Tropospheric ozone variations governed by changes in stratospheric circulation,” Nature Geoscience, vol. 7, no. 5, pp. 340–344, 2014.
P.S. I almost finished writing this a few days ago but I inadvertently hit the back button. I had to wait a few days to get back my motivation.