Pausing a autorenewal membership
In general, auto-recurring memberships are generated via an auto-recurring payment, whose mechanism depends on the payment processor, and most especially on whether the recurring payment schedule is managed by CiviCRM or the processor.

When the schedule is managed by CiviCRM, you have more direct control and could for example use some sql to just push forward the next scheduled contribution date one month (or so). You could also manually go in and edit each recurring schedule, updating the next scheduled contribution date. You'd also want to deal with the issue of the membership status - one way would be to extend the grace period, though mucking about with membership status rules can get you in weird trouble.

When the schedule is managed by the payment processor, as is the case with Stripe, (i.e. the payment processor will trigger the recurring payment according to a schedule stored on it's servers), then you'll need to depend on whether the payment processor extension gives you the ability to update that schedule. Based on one example I just looked at, that doesn't appear to be the case.

So you'd need to use the Stripe interface itself to make any changes to the schedules.