Is there a symbol for an "element"? Is there a math symbol that represents an arbitrary element?
This shouldn't be confused with the "element of" symbol $\in$.
 A: In most non-toy situations one deals with more than one set at a time, so if there WERE such a symbol, it would get confusing.  A common convention is to use the lower-case version of whatever letter used for the set.  For example, $s\in S$ or $x\in X$.
A: If we choose to look at properties instead of sets - on the grounds that, in many cases, we can conflate the two - then there is such a symbol due to Hilbert, although it's not popular outside of proof theory. However, I find it useful, at least in my own notes, and wish it had stuck around (although I wish it were denoted differently). 
The symbol is this: if $P$ is a property, then $\epsilon x P$ is a term, and the axiom governing $\epsilon P$ is $$\exists x(P(x))\implies P(\epsilon xP)$$ - that is, if there is some thing satisfying $P$, then $\epsilon xP$ is such a thing. I think the choice of "$\epsilon$" as the symbol, here, is unfortunate since it can be conflated with the relation "$\in$," especially when reading older texts; Bourbaki essentially used $\tau$ instead, which I prefer notation-wise, but they also used a logical framework I find ridiculous (and the usage of $\tau$ is much more complicated than the usage of $\epsilon$); in my own notes, I use $\tau$ in place of $\epsilon$, but never do anything else resembling what Bourbaki do, and in particular use Hilbert's rules instead of Bourbaki's for manipulating $\tau$.
This is the eponymous operator of the $\epsilon$ calculus. It was originally devised as a proof-theoretic tool for proving consistency results, and it still plays a role in proof theory (that said, for an interesting connection between the $\epsilon$-operator and computability theory, see this paper of Towsner - this followed work of Kreisel and Yang).
Now as I mentioned above, the $\epsilon$-calculus treats properties, not sets, but we can reasonably conflate the two in many circumstances, and in my  opinion it's reasonable to do so in this case (although of course the original role of the $\epsilon$-operator was to deal only with  definable sets, so this is a departure from the original intent of the symbol).
Incidentally, there's a related operator due to Frege, "$\iota$," which translates as "the unique $x$ such that" - e.g. "$\iota x(x+x=1+1)$" is a term which evaluates to $1$ in the structure $(\mathbb{N}; +, \times, 0, 1)$. (Analogously to "$\epsilon$," we need a special clause in the semantics of "$\iota$" if the relevant contidion - in this case, the existence of a unique object with the given property - is not met; in this context, a nil individual (for "undefined") is usually introduced to handle this.)
Note that the operators "$\epsilon$" and "$\iota$" are very odd, from the perspective of contemporary classical logic. Quantifiers turn formulas into formulas, function symbols turn terms into terms, and relation symbols turn terms into formulas; but these operators turn formulas into terms. Personally I find such things very cool! That said, there is a strong connection between such operators and generalized quantifiers, and they are often studied side-by-side when looking at the logic of natural language.
A: It doens't have any sense to define an element without specifying where it is. So you can pretty much choose any symbol to represent an element, but you have to specify where it is.
A: To say that $x$ is an arbitrary element of a set $S$ in the argument that follows, write 

Let $x\in S$ ...

