Which is a better nucleophile, $\ce{NH3}$ or $\ce{NH2NH2}$? For the sake of limiting scope, I want an answer for behavior in a protic solvent though non-protic solvent answers are welcomed.

My logic is that since $\ce{NH2NH2}$ has 2 lone pairs it will be a better nucleophile. I would like to know if this conclusion and reasoning is correct, or if I'm correct for the wrong reason why, or if I'm wrong, why is ammonia a better nucleophile?

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    $\begingroup$ Hydrazine is indeed a better nucleophile but I think the explanation is slightly more complex than ‘has two lone pairs’. $\endgroup$ – Jan Sep 23 '17 at 15:01

Yes, hydrazine is a better nucleophile than ammonia. This is the case in both water (a protic solvent) as well as acetonitrile (an aprotic solvent). A quantitative approach may be found in the Mayr nucleophilicity parameters (Ctrl-F "ammonia" and "hydrazine"):

$$\begin{array}{ccc} \hline \text{Nucleophile} & N\text{ parameter in }\ce{H2O} & N\text{ parameter in }\ce{MeCN} \\ \hline \ce{N2H4} & 13.46 & 16.45 \\ \ce{NH3} & 9.48 & 11.39 \\ \hline \end{array}$$

The larger, the more nucleophilic. It also makes sense that the protic solvent water reduces the nucleophilicity of both species.

For more details I suggest reading the articles cited in the database entries. The most relevant one is: J. Org. Chem. 2012, 77 (18), 8142–8155. Inside there is a good discussion of the methodology.

As for why hydrazine is a better nucleophile, the jury is still out. In general there is a phenomenon whereby nucleophiles with lone pairs on two adjacent atoms are more nucleophilic than expected, which is called the α-effect. [Note that I deliberately avoid saying that hydrazine is more nucleophilic "because of the α-effect", because that merely begs the question. The α-effect is not (yet) a theoretical explanation, but rather an empirical observation.]

I don't have the interest to summarise reviews on the α-effect (at this point in time, at least) so I will quote Mayr from the article above:

The α-effect, which has been investigated for various reactions of nucleophiles including acylations, Michael additions, SN2 reactions, and nucleophilic aromatic substitutions has been the topic of several reviews, but its origin and extent are still discussed controversially. [...] Several theories on the origin of the α-effect have emerged, which include the destabilization of the ground state by electron repulsions, the stabilization of the transition state, thermodynamic stabilization of the products, and solvent effects.

See also: Is there a general consensus on the causes of the alpha-effect? Unfortunately, that question does not have a proper answer, and I believe the direct answer to that question is "no". As Jan's comment indicates, too, the explanation must go beyond "hydrazine has two lone pairs but ammonia has one", because that alone doesn't actually explain anything.

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Yes hydrazine is a better nucleophile than ammonia (ignoring solvent effects) and this is called the "alpha effect" which states that the nucleophilicity of an atom increases if it has an adjacent atom with a lone pair of electrons.

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