First, I'm assuming you mean the ice maker attachment inside a refrigerator-freezer and not a stand-alone ice maker. This is one job that, given some basic mechanical skills, you really can do yourself if you can get the correct part needed. I've replaced bad ice makers on GE, Frigidaire, and U-Line units I've owned over the years without difficulty and in about an hour each. Each of these parts cost about $175 give or take for the parts and the labor was my own time; it was kind of enjoyable learning more how these units function.
Another advantage to DIY is that you may not need to replace the entire ice maker. Some brands sell only the complete unit, but for others you can get individual parts and replace only that which isn't working, generally at lower cost. For example, in another case of mine it was the tube coming into the freezer from the exterior water line that was bad (cracked and leaking). It turned out to be less than $4 for the part and about 5 minutes to replace. One annoying ice maker cause of failure is hard water. In one case, the teflon coating inside the ice tray started to deteriorate and left flakes in my ice. I only needed to replace the tray itself and fortunately this brand sold the parts to do so.
And lastly, check to see if you are in or just slightly out of warranty. I had an out-of-warranty ice maker go bad and I had to call the manufacturer for their parts list and diagram. In talking with them, they agreed (and even suggested) that they would sent me a new ice maker unit at no charge. Note that I was VERY nice to them in my call and that this was the first time the unit had malfunctioned in any way. Shocked me, and saved me a lot of money.
The other key component to an ice maker's functionality is the water valve unit that automatically turns the water on and off to the ice cube tray, usually located at the bottom of the refrigerator near where the house water line enters the fridge and exits to the ice maker. Again, easily replaced as a DIY project and available at online and brick-and-mortar appliance repair supply shops.
If you do go the DIY route, find your refrigerator's parts breakdown (try the manufacturer's website) to show what and how everything is connected. Be sure to turn off and disconnect water and electricity, and go at it.
Source: Experience with owned refrigerator units.
