A lot of this depends on how busy your DCs are as well as your replication topology. Expect that this will create a flood of replication traffic between your DCs - unless you have some sort of QoS in place that limits RPC traffic, you might find that the change causes a noticeable spike in LAN usage as the DCs replicate. And if the DCs themselves are already working hard then you could easily see the additional load affect all sorts of things across the enterprise - most notably, delays in connecting to your NAS. If you have multiple AD sites expect the next replication to take longer as well - and if those site links are already heavily used you might see performance decrease during the replication.
I'd do it off hours, and make sure you are able to restore from backup in case something goes wrong. Ideally, you could come up with a way to revert changes (by reapplying the old displayName) that doesn't involve restoring your entire AD from backup (this is not fun).