Does dissolving something in water change the specific heat of the solution?
If you were adding two liquids ideally, that is, assuming that the mixing occurs without any change in enthalpy - which occurs for chemically similar substances, then yes, the heat capacity of the mixture is a weighed average of the heat capacity of the components ($A$, $B$):

$$c_p = x_Ac_{pA}+(1-x_A)c_{pB}$$

However, when we are talking about dissolving solids, the situation is a little different. That is because infinite-dilution properties of a solute that derives from a solid source do not mimic the properties of the (crystalline) solid. For example, table salt ($\ce{NaCl}$) has a certain density as a powder, but this value has nothing to do with the density it acquires in solution once the ions dissociate and live freely in solution. The same goes for heat capacity, which is also a thermodynamic property.

So the answer is yes, the heat capacity of the solution does change by adding a solute and, in fact, it depends on the amount that is added. Only experimentally can you measure the exact function of concentration that defines the heat capacity of a particular solution. From a molecular perspective, it will be highly dependent on how the phenomenon of solvation develops.

As always, you can always look up the scientific literature to get the data of the thermodynamic functions of certain solutions including heat capacity.