The St Timothy concerned was not the Timothy who was a companion of St Paul but a 4th century Roman martyr. He has often been referred to as "Timotheus" (the Latin form) in English-language publications to avoid the confusion.
The revised Roman Martyrology (2001) for 22 August has St Timothy being martyred in the year 303, and being buried on the Via Ostiense. The revision deleted the following from the old edition: "He was arrested by Tarquinius, prefect of the city, and suffered a long imprisonment. When he refused to sacrifice to idols he was whipped three times, and after enduring serious tortures was at last beheaded".
The developed legend of Pope St Sylvester, which is a tendentious mediaeval confection, has the following details: St Timothy was a priest of Antioch in Syria who came to Rome in 310 and stayed with Sylvester before the latter became pope in 314. Timothy took up public preaching for fifteen months, and was then imprisoned and martyred in 311 on the orders of the prefect Tarquinius Perpenna (otherwise unknown). He was then buried in a tomb owned by a pious lady called Theon.
The Tridentine missal celebrates SS Timothy, Hippolytus of Porto and Symphorian of Autun on the same day, 22 August. See the English Wikipedia article here.
Pilgrimage shrine Edit
The tomb of the martyr, being very close to San Paolo fuori le Mura, was a shrine on the suburban pilgrimage circuit in the early Middle Ages, and seems to be the only one not associated with a set of catacombs (this is unclear). Predictably the pilgrims confused him with St Timothy the companion of St Paul, but mediaeval scholars appreciated that this popular tradition was a mistake and the error never crept into the Church's liturgical texts.
It is unknown as to when the shrine was abandoned, but the 9th century is a good guess.
The archaeologist Giovanni Battista De Rossi went looking for the shrine in 1872, exploring on the east side of the Via Ostiense near the basilica. He found an extensive burial ground, which was further uncovered when the road was widened in 1919.
This cemetery was begun in the 2nd century BC and developed over five centuries. In places, layers of tombs of differing ages were found on top of each other. Among several other hypogea, De Rossi found a single underground chamber just north of the junction with the Via delle Sette Chiese which had an access staircase. He reported pilgrim graffiti in Greek and Latin scratched on the walls of the stairwell, but no explicit mention of St Timothy. The hypogeum itself was entirely undecorated. So, De Rossi's identification of this particular tomb with the shrine of St Timothy was tentative.
According to Armellini writing at the end of the 19th century, catacomb galleries exist in the same locality but their relationship to the hypogeum is unknown. The latter might have been part of a catacomb if the galleries connect to the staircase.
The cemetery overall is sometimes called the Cimiterio di Lucina, but the preferred name nowadays is Necropoli di San Paolo. Part of it shelters under a canopy to the west of the road and north of the basilica.
According to the Pontificia Commissione di Archeologia Sacra, the hypogeum is at Via Ostiense 195/B. There is nothing to see.