Normally you shouldn't need to break up the array and create a new one – practically all RAID systems allow you to replace a disk on the fly. This is very useful for replacing failed disks, but you can use the same feature to replace a working disk with a larger one, while preserving everything else about the array, and without the filesystem even noticing that the change is happening.
(But only one at a time – you have to wait for the NAS to finish rebuilding the array before you can begin to replace the next disk.)
- Remove disk 1 and swap it with a larger disk, then allow the NAS to rebuild (resilver) it from the other disk that acted as its mirror.
- Remove disk 2 and swap it with a larger disk, wait for the NAS to rebuild.
- Swap disk 3, wait for rebuild...
- Swap disk 4, wait for rebuild...
- After all four disks are replaced, you now have a larger array.
With traditional RAID, if you have four disks, then they're not in a single RAID 1 array – they're more likely to be RAID 1+0, where two 2-disk arrays are joined together using "striping" (each mirror stores a half of every sector). While striping provides some performance, it is not very flexible – both sides of the striped array have to be identical size, which means RAID 1+0 must consist of two identically sized RAID 1 arrays.
(If you actually had a 4-disk RAID 1 array, it would only give you 25% of space due to keeping 3 mirrors. So I'm fairly sure your NAS will using RAID 1+0.)
Synology has an article about this topic, as well as a visual calculator for various array types. Here's how it illustrates the limitation of the traditional RAID, and a diagram showing how you would upgrade it (though the illustrations seem to be for RAID 5 but the idea is still the same):
After all four disks have been replaced with larger ones, the NAS should be able to make use of the increased space.
(This restriction doesn't apply to newer methods such as Btrfs, which implements not disk-level, but chunk-level mirroring/striping and allocates space differently. If this was a regular Linux system, I would really prefer Btrfs 'raid1' over actual RAID 1.)
Meanwhile Synology offers "SHR-1", which appears to be a parity mode which allows mixed-size disks and gives you 66% or more capacity, instead of just 50% for basic mirroring. As the article shows, you will immediately get extra capacity after upgrading just two disks: