Create two R functions with same name but different type of argument

As Dirk noted in the comments, this can by done by dispatching the appropriate implementation function from within the (single) exported function. The typical approach involves a switch statement and the TYPEOF macro, as shown below:

#include <Rcpp.h>

struct fallthrough {};

template <typename T>
int overloaded_impl(const T& t) {
    return -1;
}

template <>
int overloaded_impl<std::string>(const std::string& x) {
    return x.size();
}

template <>
int overloaded_impl<int>(const int& x) {
    return x * 2;
}

// [[Rcpp::export]]
int overloaded(SEXP x) {
    switch (TYPEOF(x)) {
        case INTSXP: {
            return overloaded_impl<int>(INTEGER(x)[0]);
        }
        case REALSXP: {
            return overloaded_impl<int>((int)(REAL(x)[0]));
        }
        case STRSXP: {
            std::string tmp = CHAR(STRING_ELT(x, 0));
            return overloaded_impl<std::string>(tmp);
        }
        default: {
            Rcpp::warning("Unmatched SEXPTYPE!");
            return overloaded_impl<fallthrough>(fallthrough());
        }
    }
    return -1; // not reached
}

/*** R

overloaded("a string")
#[1] 8

overloaded(10L)
#[1] 20

overloaded(10)
#[1] 20

overloaded(TRUE)
#[1] -1
#Warning message:
#In overloaded(TRUE) : Unmatched SEXPTYPE!

overloaded(2 + 2i)
#[1] -1
#Warning message:
#In overloaded(2 + (0+2i)) : Unmatched SEXPTYPE!

*/ 

---

The case: REALSXP is just there because R defaults to numeric instead of integer, e.g. without it you would have:

overloaded(10)
#[1] -1
#Warning message:
#In overloaded(10) : Unmatched SEXPTYPE! 

---

A variation of this strategy is to create a wrapper class housing a variant object, wherein the switch-based type deduction logic is moved into the constructor, and method dispatch is carried out by applying the visitor pattern. This is not really justified for the simple example above, but in situations where you have several different functions that may be called on the object, it can save you from a lot of code duplication, as the switch(TYPEOF(x)) {...} block does not need to exist in each function. Here's an example where I've done this on a larger scale with the Boost C++ libraries, courtesy of the BH package.

At any rate, we could rewrite the original example using the variant / visitor technique as follows:

// [[Rcpp::depends(BH)]]
#include <Rcpp.h>
#include <boost/variant.hpp>

class variant {
private:
    struct fallthrough {};
    typedef boost::variant<
        int,
        std::string,
        fallthrough
    > variant_t;

    variant_t v;

    struct overloaded_visitor : public boost::static_visitor<int> {
        int operator()(const std::string& x) const {
            return x.size();
        }

        int operator()(const int& x) const {
            return x * 2;
        }

        template <typename T>
        int operator()(const T&) const {
            return -1;
        } 
    };

public:
    variant(SEXP x) 
    {
        switch (TYPEOF(x)) {
            case INTSXP: {
                v = variant_t(INTEGER(x)[0]);
                break;
            }
            case REALSXP: {
                v = variant_t((int)(REAL(x)[0]));
                break;
            }
            case STRSXP: {
                std::string tmp = CHAR(STRING_ELT(x, 0));
                v = variant_t(tmp);
                break;
            }
            default: {
                Rcpp::warning("Unmatched SEXPTYPE!");
                v = variant_t(fallthrough());
                break;
            }
        }
    }

    int overloaded() const {
        return boost::apply_visitor(overloaded_visitor(), v);
    }
};


// [[Rcpp::export]]
int overloaded(SEXP x) {
    return variant(x).overloaded();
}

/*** R

overloaded("a string")
#[1] 8

overloaded(10L)
#[1] 20

overloaded(12)
#[1] 24

overloaded(FALSE)
#[1] -1
#Warning messages:
#In overloaded(FALSE) : Unmatched SEXPTYPE!

overloaded(2 + 2i)
#[1] -1
#Warning messages:
#In overloaded(2 + (0+2i)) : Unmatched SEXPTYPE!

*/

The RcppExport is a simple
#define RcppExport extern "C"
Hence, myFunction has a C-like naming convention. Therefore, it can not be overloaded, because you need C++ style of names for this case.