Cause would probably mean a material mistake made by the law firm. Missing a filing date, not returning your telephone calls, or improper billing may amount to "cause," An "attorney lien" usually means (at least in the state I practice) that the attorney will refuse to release your file to you. This occurs when you do not pay your bill or replenish a retainer as specified in your retainer agreement.
My retainers have such language. I always explain that discharge for cause means the attorney has committed misconduct of some sort. The most extreme example would be stealing the client's money. A more common example would be overcharging the client, or neglecting the client's matter. An attorney discharged for cause forfeits his lien. There are two kinds of liens in NY. One permits the attorney to be paid for services rendered when discharged or to be paid at the end of the case if there was a contingency contract. The other lien permits the attorney to retain the client's file until paid.
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It basically means the lawyer did something improper - not just had a disagreement with the client.
I am a former federal and State prosecutor and now handle criminal defense and personal injury/civil rights cases. Feel free to check out my web site and contact me at (212) 385-8015 or via email at Eric@RothsteinLawNY.com. The above answer is for informational purposes only and not meant as legal advice.