Well, you have several options to assist you in locating your missing relative. You didn't mention if your cousin was in the United States or in a foreign country, so I will assume the answer is the United States. You can simply hire a licensed private investigator who will conduct a search based upon what information you may have available. You will be asked to pay a retainer of course, sign a contract and agree to pay an hourly rate in addition to expenses. Unfortunately, if the person you hire fails to locate your cousin, you will still be financially obligated according to the contract.
On the other hand, depending on the circumstances such as your cousin is going to inherit something from the estate and that value, there are a few really good forensic genealogists who specialize in exactly this work. They are far more knowledgeable and adept at locating heirs to estates than a private investigator who is more-or-less, a jack of all investigative trades and not an heir specialist.
When you retain a forensic genealogist, usually you will not upfront any fees or money, as the genealogists compensation will come directly from the heir you need found. By that I mean that the genealogist will find your cousin and enter into a contingent fee agreement, whereby he will receive a small percentage of what your cousin inherits for the investigative efforts. If they cannot find your cousin or they do and your cousin won't sign a compensation agreement, usually the forensic genealogist would ask the Personal Representative for some type of reasonable compensation which may or may not be granted.
When working with a forensic genealogist always be sure to verify their experience and integrity whcich only occurs after many, many years of locating missing heirs.