Any slave could get manumission in theory. It was common and selective and used as a tool to promote good behaviour. My prof described it as a lottery, each day the slave behaved well he got a 'ticket' to this lottery. But why would the master free his slaves? He spent considerable capital on them and freeing them means he's giving up that investment. Furthermore manumission carried a cost - there was a 5% tax on it starting in 357 BCE (although we're not sure what the amount was based on).

Masters could give their slaves a peculium which they could invest and they'd be allowed to keep the profit. A slave could save up this profit and use it to buy their freedom. In essence, the slave paid for his replacement. And even being freed, the former slave was not completely free of their former master, they were expected to become his clients and were bound by the ties of the Roman patron-client relationships. Failure to follow this could have their freed status revoked according to Augustan law. The other side is that a freedman would have no assets - which means he'd be working again for his master out of necessity, but with a wage at least. So manumission creates a new client for the master, may have generated sufficient money to pay for their replacement, and would likely be working again for the master (albeit with a wage) so their skills wouldn't be lost. These factors (clients, investing money, retaining skills) suggest a professional slave, one with specialized training and education. So a craftsman, a secretary, a scribe rather than a field hand.

Lastly there is the personal factor. A slave working in the household of the master has plenty of opportunities to build a relationship, become a person rather than just a tool with a name. A farm slave working on an estate with an absentee landlord will almost certainly never become known to the master other than as an account entry. That said, even if the master is very highly disposed towards the slave, there is still no guarantee of freedom. Seneca talks about his favourite slave still being servile in his household despite his many years of service. Cicero didn't free his faithful slave Tiro until he was well into his 30s.

So the short answer to your son is that the farm slave could get manumitted but almost certainly never was.