In the poem "On My First Son" by Ben Jonson, what type of diction does he use?

I'm not quite sure. It's an odd mix of the absolutely colloquial and personal, and a slightly arch, grander style.

The first four lines, addressing the son are particularly colloquial:

Farewell, thou child of my right hand, and joy ;
    My sin was too much hope of thee, lov'd boy.
Seven years thou wert lent to me, and I thee pay,
    Exacted by thy fate, on the just day.

He addresses the child as "lov'd boy", and in the more colloquial "thou" rather than the more formal "you". Yet then the poem becomes a little more abstract, addressing bigger themes in a more formal, rhetorical fashion:

Oh, could I lose all father now ! For why
    Will man lament the state he should envy?
To have so soon 'scaped world's and flesh's rage,
    And if no other misery, yet age !

'World's and flesh's rage' sounds to me at once more grand and more formal. It becomes, if you like, more consciously a piece of poetry: speaking as 'man', rather than as one person. It heightens itself: ready for its conclusion:

Rest in soft peace, and, asked, say, Here doth lie
    Ben Jonson his best piece of poetry.
For whose sake henceforth all his vows be such
    As what he loves may never like too much.

At once, Jonson teaches his young child a new phrase "Here doth lie...", and formalises his child and his life into a formal poem. And then, the final couplet addresses itself in a more poetic way: "all his vows be such /As what he loves...", with a self conscious riddling phrase at the end.

So the answer is ultimately formal diction. But it begins more colloquially - and I reckon you could argue it either way.

Hope it helps!