How to state the adequacy of an encoding of lambda calculus in itself?
As pointed out by others, the obvious definition of an "adequate" coding is that it is equitranslatable with any standard one.  The question is therefore to characterize such codings in terms of more elementary properties.

(Historical note.  Smullyan studied this question in the context of combinatory logic.  When I was a student, Henk Barendregt suggested Smullyan's conjectures to me as a research problem -- which led to my first scientific publication.)  See

http://drops.dagstuhl.de/opus/volltexte/2011/3249/
$\newcommand{\code}[1]{\overline{#1}}$

To summarize, given a map $\code{\cdot} : \Lambda \to \Lambda$, we consider whether there exist combinators satisfying certain properties.  The most significant ones are:


$A\, \code{M}\, \code{N} = \code{MN}$
$B\, \code{M} = \code{\code{M}}$
$P_i\, \code{M_0 M_1} = \code{M_i},\quad i \in \{0,1\}$
$Z_b\, \code{M} = \begin{cases} \lambda x y.x &amp;M \equiv b \in \{I,K,S\}\\
\lambda xy.y &amp;otherwise
\end{cases}$
$\Delta\, \code{M}\, \code{N} = \begin{cases}
\lambda x y. x &amp;M \equiv N\\
\lambda x y.y &amp;otherwise
\end{cases}$
$U\, \code{M} = \code{M}^s$, where $\code{\cdot}^s$ is some standard coding
$U^{-1}\, \code{M}^s = \code{M}$


It is easy to see that any adequate coding has these properties.  The main result of the paper (Corollary 14) is that, for an adequate coding, one of the following suffices:


$A + \Delta$;
$U^{-1} + \Delta$;
$U^{-1} + P_i + Z_b$;
$P_i + Z_b + $ "the range of $\code{\cdot}$ is contained in a recursively enumerable set and there is a combinator that decides whether a given element of this set is in the range or not".