Commodus' name right

Romans only used their full formal names on certain occasions. At other times, they would use just their given name. Commodus' given name was just that. His full name was Lucius Aelius Aurelius Commodus. And this is further complicated by the extensive practice of adoption among patrician Romans. For example, Marcua Aurelius' birth name was Marcus Annius Verus. Commodus (Lucius Aelius Aurelius Commodus), 161–192, Roman emperor (180–192), son and successor of Marcus Aurelius. In 180, reversing his father's foreign policy, he concluded peace with the German and the Sarmatian tribes and returned to his licentious pleasures in Rome. There he vaunted his strength in gladiatorial combats and decreed that he should be worshiped as Hercules Romanus. He changed his own name to Marcus Commodus Antoninus and wanted to rename the city of Rome after himself. Many plots to assassinate him failed, but eventually, on the order of his advisers, he was strangled by a wrestler. Pertinax succeeded him.

Commodus' full name.

Commodus' full name is Marcus Aurelius Commodus Antoninus. He was popularly known as just Commodus. Most Roman (male) names consist of four or five names, but are generally known by one or two. Also, the family name was Antoninus, because he was one of, and in fact the last of, the Antonine dynasty. It doesn't always follow that a daughters name matches her fathers name, but it is, however, very common. It could have been she changed her name to match her husbands name (late husband at the time of the film) which was Lucius Verus.

Lucilla a possible name

You are correct that Roman women did not in general have their own names. However, They did not solely use their father's names. They sometimes used feminine forms of their husbands' names as well. Lucilla, who was married to a man named Lucius, could have been called Lucilla for that reason.

Named after her mother, perhaps?

Daughters were sometimes named after their mother. Livia, wife of the emperor Augustus, had a daughter from her previous marriage. This daughter was also called Livia, but was known as Livilla, or 'Little Livia'. So, Lucilla's mother may have been Lucia.

Adoptive emperor's names

In the case of the adoptive emperors dynasty, it's quite confusing to determine the family relation from the names. The emperor known as Marcus Aurelius was born M. Annius Verus on 26/04/121, the son of M. Annius Verus and Domitia Lucilla. When the emperor Hadrian (P. Aelius Hadrianus) had to think about who would succeed him, his first choice was a L. Ceionius Commodus, who died in 132. His second choice was a toss-up between young Annius and the able senator Titus Aurelius Boionius Arrius Antoninus - he adopted the latter and under the condition that he in turn would adopt both Marcus Annius, who thus became known as M. Aelius Aurelius Verus, and L. Ceionius Commodus, the son of his earlier choice, who became known as Lucius Aelius Aurelius Verus. Titus Aurelius himself went down in history as the emperor Antoninus Pius. The 'family' name of all these emperors is Aelius. That means that if M. Aurelius had had a daughter, her given name would have been Aelia. But if he had had more, he would have probably given her an additional name to distinguish her from her sister(s). In classical, i.e. Republican times, her cognomen would have been Minor or Tertia or something like that, but in the days of the Empire, people weren't that strict anymore. As Lucilla was Marcus Aurelius' mother's cognomen, it could easily have been given to (one of) her granddaughter(s). Marcus Aurelius' son Commodus' full name was M. Aurelius Commodus Antoninus, the name Commodus no doubt having derived from and given in memory of his father's co-emperor Lucius Verus, who died in 169.

Roman girls would take their fathers first names. EX: father - Cornelius, daughter - Cornelia. The first son would normally take the father's name, but every son after that would be named by what number son they were. EX: the 6th son in a family would be named Sextus. Commodus' original name was Lucius Aelius Aurelius Commodus, but later changed it to Marcus Aurelius Commodus Antoninus. The daughter's name was actually Lucilla.