The NIV translation is somewhat misleading, I think. The literal Greek uses a relative pronoun ("which") and not a demonstrative ("this"), which would imply that it was the famine and not Agabus' saying that took place during the time specified.
Claudius ruled between 41 and 54 AD.
One possibility is that Agabus was predicting a famine that Tacitus later described his Annals:
Several prodigies occurred in that year. Birds of evil omen perched on
the Capitol; houses were thrown down by frequent shocks of earthquake,
and as the panic spread, all the weak were trodden down in the hurry
and confusion of the crowd. Scanty crops too, and consequent famine
were regarded as a token of calamity.
- Annals 12:43
"That year" seems to be the year that a certain Junius Lupus was charged with treason (12:42), which Barbara Levick dates to 51 AD in her biography of Claudius. This would have been around 10 years after Agabus prophesy (which scholars date to around 42 AD).
Another possibility is that he is referring to a famine that took place specifically in Judea at the time of the Procurator Tieberius Alexander, who ruled between 46 AD to 48 AD, described by Josephus (Antiquities XX.V.2). The Greek text that the NIV translates as "over the entire Roman world" is more literally read, "over all the 'inhabited world' [οἰκουμένην], which was certainly used to mean the Roman Empire (i.e. the 'inhabited world' as someone in the Roman Empire might perceive it); but it might also perhaps refer to just the kingdom of Judea.
Endicott speculates that the famine referred to in Acts could also be one referred to in Suetonius' Life of Claudius but I could not find any reference to a famine in that work.