When converting zinc to copper, what does the hydrochloric acid do?

Zinc is more electronegative than copper; therefore, a redox reaction would not occur between these two metals as presented here.  However, zinc will oxidize in the presence of acid to give H2(g) as a byproduct.  At the same time, H2(g) will react with Cu2+(aq) to form Cu(s) and H+(aq) again, thus you can think of HCl(aq) as acting like a catalyst for this reaction (provided H2(g) is not lost to the surroundings).  The reactions are these:

Zn(s) + 2 H+(aq) ---> Zn2+(aq) + H2(g)
Cu2+(aq) + H2(g) ---> Cu(s) + 2 H+(aq)

Overall reaction:
Zn(s) + Cu2+(aq) ---> Zn2+(aq) + Cu(s)

As mentioned above, this redox reaction would not occur if it were not due to the intervention of H+/H2.  You can rationalize this based on the activity series of the metals:
http://en.wikipedia.org/wiki/Activity_series

Cheers.