On what basis do Popes choose their papal name? Apologies if there is not a good answer to this but I was wondering how papal names are chosen. Is there a specific tendency for taking a papal name that is associated with a tradition or strain of thought? For instance does the name Benedict imply that you are from a particular subset of orders, does Pius imply that you have a particularly high Christology? Or are the names essentially down to the Pope in question's relatively arbitrary choice? This question could I suppose be reframed as do Papal names carry any particular connotations?
It varies.

For example, Benedict XVI chose his name after St. Benedict and Pope Benedict XIV.

Pope Pius XII likely chose "Pius" because he is in the episcopal lineage of Pope St. Pius X.

John Paul I and II chose theirs after the preceding John XXIII and Paul VI.

Christ chose St. Peter's for him. ☺

Other than that, yes, "the names essentially [come] down to the Pope in question's relatively arbitrary choice". After they are elected, a cardinal asks them which name they choose.