When a person is seeking to change their name due to any reasons other than a divorce, the family courts are the place to seek relief. The process itself is fairly simple and involves filing a Petition for Name Change, together with electronic fingerprints. If there are no problems with the background check, the petition goes before a judge in family court that signs a final judgment effectively changing a person's name. Said final judgment can then be delivered to all parties affected such as the DMV, USCIS for passport, etc.
The Florida Statute that deals with this is FS 68.01:
68.07 Change of name.—
(1) Chancery courts have jurisdiction to change the name of any person residing in this state on petition of the person filed in the county in which he or she resides. (2)(a) Before the court hearing on a petition for a name change, the petitioner must have fingerprints submitted for a state and national criminal history records check, except if a former name is being restored. Fingerprints for the petitioner shall be taken in a manner approved by the Department of Law Enforcement and shall be submitted electronically to the department for state processing for a criminal history records check. The department shall submit the fingerprints to the Federal Bureau of Investigation for national processing. The department shall submit the results of the state and national records check to the clerk of the court. The court shall consider the results in reviewing the information contained in the petition and evaluating whether to grant the petition. (b) When a petition is filed which requires a criminal history records check, the clerk of the court shall instruct the petitioner on the process for having fingerprints taken and submitted, including providing information on law enforcement agencies or service providers authorized to submit fingerprints electronically to the Department of Law Enforcement. (c) The cost of processing fingerprints and conducting the state and national criminal history records check required under this subsection shall be borne by the petitioner for the name change or by the parent or guardian of a minor for whom a name change is being sought. (3) Each petition shall be verified and show: (a) That the petitioner is a bona fide resident of and domiciled in the county where the change of name is sought. (b) If known, the date and place of birth of the petitioner, the petitioner’s father’s name, the petitioner’s mother’s maiden name, and where the petitioner has resided since birth. (c) If the petitioner is married, the name of the petitioner’s spouse and if the petitioner has children, the names and ages of each and where they reside. (d) If the petitioner’s name has previously been changed and when and where and by what court. (e) The petitioner’s occupation and where the petitioner is employed and has been employed for 5 years next preceding the filing of the petition. If the petitioner owns and operates a business, the name and place of it shall be stated and the petitioner’s connection therewith and how long the petitioner has been identified with that business. If the petitioner is in a profession, the profession shall be stated, where the petitioner has practiced the profession, and if a graduate of a school or schools, the name or names thereof, date of graduation, and degrees received. (f) Whether the petitioner has been generally known or called by any other names and if so, by what names and where. (g) Whether the petitioner has ever been adjudicated a bankrupt and if so, where and when. (h) Whether the petitioner has ever been arrested for or charged with, pled guilty or nolo contendere to, or been found to have committed a criminal offense, regardless of adjudication, and if so, when and where. (i) Whether any money judgment has ever been entered against the petitioner and if so, the name of the judgment creditor, the amount and date thereof, the court by which entered, and whether the judgment has been satisfied. (j) That the petition is filed for no ulterior or illegal purpose and granting it will not in any manner invade the property rights of others, whether partnership, patent, good will, privacy, trademark, or otherwise. (k) That the petitioner’s civil rights have never been suspended or, if the petitioner’s civil rights have been suspended, that full restoration of civil rights has occurred. (4) The hearing on a petition for restoring a former name may be held immediately after it is filed. The hearing on any other petition for a name change may be held immediately after the clerk receives the results of the criminal history records check. (5) On filing the final judgment, the clerk of the court shall, if the birth occurred in this state, send a report of the judgment to the Office of Vital Statistics of the Department of Health on a form to be furnished by the department. The form must contain sufficient information to identify the original birth certificate of the person, the new name, and the file number of the judgment. This report shall be filed by the department with respect to a person born in this state and shall become a part of the vital statistics of this state. With respect to a person born in another state, the clerk of the court shall provide the petitioner with a certified copy of the final judgment. (6) The clerk of the court must, upon the filing of the final judgment, send a report of the judgment to the Department of Law Enforcement on a form to be furnished by that department. The Department of Law Enforcement must send a copy of the report to the Department of Highway Safety and Motor Vehicles, which may be delivered by electronic transmission. The report must contain sufficient information to identify the petitioner, including the results of the criminal history records check if applicable, the new name of the petitioner, and the file number of the judgment. Any information retained by the Department of Law Enforcement and the Department of Highway Safety and Motor Vehicles may be revised or supplemented by said departments to reflect changes made by the final judgment. With respect to a person convicted of a felony in another state or of a federal offense, the Department of Law Enforcement must send the report to the respective state’s office of law enforcement records or to the office of the Federal Bureau of Investigation. The Department of Law Enforcement may forward the report to any other law enforcement agency it believes may retain information related to the petitioner. (7) A husband and wife and minor children may join in one petition for change of name and the petition must show the facts required of a petitioner as to the husband and wife and the names of the minor children may be changed at the discretion of the court. (8) When only one parent petitions for a change of name of a minor child, process shall be served on the other parent and proof of such service shall be filed in the cause; however, if the other parent is a nonresident, constructive notice of the petition may be given pursuant to chapter 49, and proof of publication shall be filed in the cause without the necessity of recordation. (9) This section does not apply to any change of name in proceedings for dissolution of marriage or for adoption of children.