Do mitochondria contain the genes to specify themselves?
The answer is a bit more complicated than that. Mitochondria contain their own genome called mitochondrial DNA (mtDNA), encoding 13 proteins that are part of respiratory complexes I, III, IV, and V, 22 transfer RNAs (tRNAs), and two ribosomal RNAs (rRNAs). The separate tRNAs and rRNAs are necessary because the mitochondrial genome uses a slightly different genetic code than the nuclear genome.

So, the mitochondria contain nearly all of the genes necessary to construct the respiratory chain, as well as the machinery to translate the mRNA into protein. However, it does not contain any of the genes necessary to actually replicate the mtDNA (DNA polymerases, etc.) or construct the mitochondrion itself - synthesize the components of the different membranes, arrange the membranes in their required conformation, chaperones to assemble the complexes of the respiratory chain, etc. - these are all encoded in the nuclear DNA.