I doubt the solution become colder. Solvation of salt (NaCl) is an exothermic process. In fact, adding salt lowers the melting point of ice, hence more ice is formed, thus it seemed as if the temperature of the solution drops.
Salt
lowers the melting point of ice because it increases the entropy of the aqueous phase so the (Delta)S from aqueous to solid becomes higher. Since for ice melting at its melting point (Delta)G = 0 and therefore, T = - (Delta)H / (Delta)S, a larger (Delta)S results in a smaller T.
Therefore, since the temperature of an ice-water mixture at thermal equilibrium will be the melting temperature of ice, adding salt to an ice-water mixture will lower the temperature of an ice-water mixture. So therefore, in the sittuation henry321 describes, the temperature of the ice-water mixture which forms when ice melts will be colder than an ice-water mixture without salt.