if he is entitled to any attorney fee lien it would be for the reasonable amount of hours he has spent on the file to the date he was discharged. However, if he is quitting your case and moving to withdraw he may not have a right to a lien. Talk to a lawyer in your area.
In addition to Attorney Sullivan's response, I suggest you re-post under Chicago, IL, since Illinois law and the Illinois rules regarding attorneys would apply.
The information presented here is general in nature and is not intended, nor should be construed, as legal advice. This posting does not create any attorney-client relationship with the author (who is only admitted to practice law in the State of California). For specific advice about your particular situation, consult your own attorney.
A client has a right to discharge an attorney at any time for any reason. If the attorney is discharged "for cause" or if thre is good cause to fire him he forfeits his fee. It may not matter if the attorney is withdrawing or not. The client can always say, you can't quit because I am firing you. Whether that works is depends on the facts of the case and how the judge rules on the motion to withdraw. What constitutes "for cause" is not always a black and white question. If an attorney is not discharged for cause he is entitled to be paid for his time and expenses or to receive a fair share of the contingent fee at the end of the case. This is only a very basic overview of the legal issues involved and I agree with the other lawyers that your cousin needs to get advice from an Illinois lawyer because the outgoing attorney may have other rights. In New York the outgoing attorney has a right to retain the file until his fee is decided. This may not apply in Illinois. Fee disputes can get very ugly and can prejudice the client's case. It is always wise to get fee issues resolved at the time of substitution if that is possible.