Copper and hydrochloric acid
If pure hydrochloric acid is added to a sample of pure copper, there is essentially no reaction; the solution should not turn blue.  

At a guess, you're seeing the copper turn pink because it has a significant oxide layer that is reacting with the acid, forming copper (II) chloride and leaving bare the pinkish metal.  The solution could be pale blue or pale green, and the time delay of the color would be a result of the amount of time required for the reaction to occur.  Note that if this is the case, you would not expect to see any bubbling on the copper, as copper (II) oxide will react with hydrochloric acid to form copper (II) chloride and water.

As Ivan Neretin pointed out in comments, large pieces of copper will react with hydrochloric acid over a period of several hours to days in the presence of air.  Permeakra added that a fine copper powder will react in minutes with a solution of hydrochloric acid in the presence of air.  In both cases it is the addition of oxygen that causes the copper to react, and in both cases the solution will be a greenish yellow color due to the excess of hydrochloric acid required.

As an example, a student of mine recently asked about this with regard to a penny.  We cut open a modern US penny (95% zinc, 5% copper by mass) to expose the zinc core, and placed it in 200mL of 6M HCl.  The oxidation layer on the penny was gone almost immediatley after it entered the HCl solution.  The zinc reacted quickly and by the next day the penny had been converted into a hollow copper shell.  The shell sat in the remaining HCl solution (still 5.7M) over the weekend with no visible reaction but the solution slowly turned green and the copper dissolved completely by the following Friday.

If your solution is turning pink, then I'm not entirely sure what's going on.