Unfortunately, there is not a 'best' fucntioanl to perform NMR calculation in Gaussian. All depends on the systtem that you want to study. So, knowing the system you could decide and try some functional and see which is the most suitable for it.
Furthermore, the interesting thing is the Keyword NMR. Here you can fin a couple of examples.
www.gaussian.com/g_tech/g_ur/k_nmr.htm
http://web.chem.ucsb.edu/~kalju/chem112L/public/nmr_split.html
B3LYP
The method of choice is Density Functional Theory with the B3LYP functional, since it is well-studied and provides superb trade-off between accuracy and computational speed for the systems studied.
As always and as Melchor suggested, it depends on the system.
First: make sure that you have a functional that is able to accurately predict the geometry of the system. If you have a rather standard system with no strange/rare features, start with B3LYP and a moderate, double-zeta basis set (with diuffusion functions if needed).
Second: do GIAO NMR for the optimized structure, at least in the 6-31G(d) basis set (or just in case and if you can afford it in 6-311G+(d,p)). As long as it's an organic molecule, the prediction should be very good.
Third: if you don't like the geometry and/or are in need of newer functionals (e.g. for a system with important pi-cation interactions, for which you'll need M06 class of functionals or those with D3 correction) - beware, however, that the number of NMR validations studies for these functionals is much lower and there may be some surprising results.
The hybrid and meta hydrid functionals give reliable results. The more important point is to use large and appropriate basis sets
Check out http://cheshirenmr.info/ ,especially the section "recommendations" and the references therein. They list several good DFT method/basis set combinations, including linear scaling factors to better match the experimental data.
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