User-defined random variable

I would like to know whether it is possible with R to define a
discrete random variable different from the ones already defined
inside R and generate random numbers from that user-defined
distribution.

Yes. One generic way is to specify the quantile function (as in
qpois() etc.) and do qfun(runif(N)).

If the support discrete but also finite, you can also use sample(), e.g.

 sample(myset, N, replace = TRUE, prob = myprob)

one can also use our R package "distr" to generate discrete random
variables. The subsequent code provides a function which generates an
object of class "DiscreteDistribution" based on a finite support "supp".
If "prob" is missing all elements in "supp" are equally weighted.