The Leibnizian cosmological argument (LCA) goes like this:
1. Everything that exists has an explanation of its existence, either in the necessity of its own nature or in an external cause. (Premise)
2. If the sum total of contingent entities C has an explanation of its existence, that explanation is a necessary, eternal and very powerful entity N. (Premise)
3. C exists. (Premise)
4. Hence, C has an explanation of its existence. (From 1 and 3)
5. Therefore, the explanation of C is N. (From 2 and 4)
Hume objects that once each member of a regress of causes is accounted for, then it is superfluous to ask for an explanation of the regress itself. Maybe there is an infinite regress of contingent entities, each causing the next. Thus, he rejects premise (2).*
Now, the traditional proponent of the LCA will point out, quite rightly I think, that the infinite regress (if one is even possible) itself is contingent. There didn't have to be any contingently existing entities, and so there did not have to be an infinite regress of contingent causes. Hume's proposed solution doesn't result in an explanation for the infinite regress.
Nevertheless, there's an additional argument at the disposal of Leibniz, and I'm surprised I never see it brought up in defenses of the LCA. What I'm talking about is the one-to-one correspondence between potentialities and an infinite regress. Thomas Aquinas sums it up quite nicely in the Third Way:
6. Something cannot come from nothing. (Premise)
7. Something presently exists. (Premise)
8. Either everything that exists is contingent, or else there exists at least one necessary entity. (Premise)
9. The past is infinite. (Assumption)
10. Given infinite time, all potentialities will have been actualized. (Premise)
11. The concurrent non-existence of all contingent entities is a potentiality. (Premise)
12. Hence, there was a time in the past at which nothing contingent existed. (From 9 - 11)
13. Therefore, at least one necessary entity exists. (Implied by 6, 8 and 12)
In short, since there was a time in the past at which nothing contingent existed, it is either the case that something necessary existed at that time (and therefore must exist at all times) or that nothing at all existed. The problem with the latter is premise (6) of the argument. If there were ever a past time at which nothing existed, then nothing would exist even now, for being cannot arise from non-being. Given that something presently exists, it follows that at least one necessary entity exists.
As a result, Hume's objection is further weakened. There are good reasons to believe the infinite regress he speaks of is itself contingent, and secondly, even if it isn't contingent, there would still have been a past time at which nothing contingent existed. Either way, there's no escaping the conclusion that N exists.
*Hume sort of rejects (1) as well, but also accepts it as an indispensable element of human thought.