Yup. It was a thing. It's a practice called evocatio - calling out. Here's Mary Beard from the OCD:

A ritual by which, in the course of a war, a Roman general would attempt to deprive the enemy of divine protection, by formally offering their protecting deity a new home and cult at Rome. The clearest recorded case is the evocation of *Juno Regina from the Etruscan city of *Veii in 396 bce (Livy, 5. 21 ff.); the ritual led to the establishment of her cult on the *Aventine hill in Rome. There has been some debate as to how long the ritual continued to be practised, and (in particular) whether the record of the evocation of Juno from Carthage in 146 bce (Serv. on Aen. 12. 841) is anything more than antiquarian invention. The discovery of an inscription at Isaura Vetus (in modern Turkey), apparently recording an evocatio in c.75 bce, suggests that the ritual survived at least to the late republic. There are, however, changes from earlier practice: in 75 bce the deity seems to have been offered a home not in Rome, but in provincial territory.

That siege of Veii is what your friend is thinking of. We don't know if that's something that really happened, or if Romans projected a contemporary tradition into their past. 396 is pretty early. A common point for establishing historicity is the Gallic sack of Rome in 390 - the idea being that after 390 we can reasonably assume some consistent record keeping that would have been available to later historians, while it's more difficult to imagine records surviving from before the sack. 396 is just a bit too early, if you buy that argument.

Romans did something similar when suffering from a plague in 293. Livy's account is lost, but we have a summary:

When the people suffered from a plague, envoys were sent to bring a statue of Aesculapius from Epidaurus to Rome. They brought with them a snake that had joined them in the ship, and which no doubt was a manifestation of the god; from the ship, it went to the island in the Tiber, to the place where the temple of Aesculapius has been erected.

Here's a link to the Veii story in Livy. Scroll down to paragraph 5.21.