We know that Roman men often named sons after them, and daughters were usually given the feminine form of their father's name. But are there any records or evidence that a Roman man was ever given the masculine version of his mother's name?
Yes, they were, but not until the Imperial period. A specific example is this one:
The practice of Constantinius Aequalis and Pacatia Servanda is typical of the later first or second century. The couple had three sons. They named the eldest Constantinius Servatus, his cognomen a development of his mother's. The second they named Constantinius Aequalis after his father and the third Constantinius Constans.
Source: Benet Salway, 'What's in a Name? A Survey of Roman Onomastic Practice from c. 700 B.C. to A.D. 700'. In 'The Journal of Roman Studies Vol. 84 (1994), pp. 124-145'
One reason for using the mother's name was if she was from a prestigious or wealthy family, and this seems to have been the case with the more famous example of the emperor Vespasian (Titus Flavius Vespasianus) whose mother, Vespasia Polla, was from a more notable family than his father, Titus Flavius Sabinus. Vespasian's father
Sabinus married up.... Vespasia Polla came from a family eminent at Nursia, an Umbrian town abutting on Sabine territory, and her father Pollio, as well as holding the post of Camp Prefect, served three terms as military tribune, a post for intending senators as well as equites. Suetonius notes a place 6 miles from Nursia on the road to Spoletium known as ‘Vespasiae’ which showed many sepulchral monuments of her family and proved its distinction.
Source: Barbara Levick, 'Vespasian' (1999)
In this case (unlike the first example cited above), Vespasian was the second-born son:
The imperial family of Vespasianus illustrates another fashion in naming during this period - and that was the practice of giving a second-born son a name derived from the mother's side of the family, to distinguish him from his elder brother who would be given the father's name. The name of the emperor Vespasianus had been derived from his mother's name, Vespasia, rather than from his father's cognomen of Sabinus (which had been given to Vespasian's elder brother, while both brothers bore the same praenomen of "Titus", after their father).
Another emperor, Antoninus Pius (ruled 138 to 161 AD, full name: Titus Aurelius Fulvus Boionius Arrius Antoninus), also bore the male version of his mother's name (Arria Fadilla) before he became emperor. As emperor, though, he was Imperator Caesar Titus Aelius Hadrianus Antoninus Augustus Pius ('Arrius' was dropped). Also, the birth name of Severus Alexander (ruled 222 to 235 AD) was Marcus Julius Gessius Bassianus Alexianus, the Julius seemingly coming from his mother Julia Avita Mamaea but, as with Antoninus Pius, this was dropped (Caesar Marcus Aurelius Alexander) when he was adopted and when he was emperor (Imperator Caesar Marcus Aurelius Severus Alexander Augustus). For the most part, though, emperors do not appear to have borne the male version of their mother's name (none of the 11 emperors between Antoninus Pius and Severus Alexander seem to have done).
One area of potential confusion is that the mother's family name could also be taken by adopting the maternal grandfather's name rather than using it because it was the name of the mother. How common this was exactly is hard to establish as, during the Imperial period, naming became far more complex than it had been during the Republican period, thus making it difficult at times to establish the true origin of some names. Salway's examples of Gaius Plinius Caecilius Secundus (Pliny the Younger) and the senator Gaius Bruttius Praesens Lucius Fulvius Rusticus illustrate just some of the complexity evident in Roman naming. Note that in neither case did they take their mother's name.
The binary nomenclature resulting from maternal inheritance and testamentary adoption is largely indistinguishable without additional biographical information. For instance, despite their similar formulation, the names of Praesens and Pliny are the result of different situations. The former, the natural son of a L. Bruttius, has appended the nomina of his maternal grandfather, L. Fulvius Rusticus, to his paternal nomina. Pliny on the other hand was born P. Caecilius Secundus, the son of L. Caecilius Cilo and a Plinia, and was adopted in A.D. 79 by the will of his maternal uncle, C. Plinius Secundus. This testamentary adoption achieved its objective admirably since, although originally a Caecilius, the testator's heir has been remembered as a Plinius.
So, for example, in the case of Pliny the Younger, one might - without additional information concerning his birth name - assume that he bore the name Plinius because of his mother (Plinia Marcella), whereas in fact Plinius came from his mother's brother, Pliny the Elder, who adopted him.