I don't know your situation but are you and your mother under the same roof? If yes, you have a few options ranging from drives you attached to your computers independent of one another (external drive per each computer), you may also consider NAS (storage on the network) and if you get another router with a good reputation on their USB3 port, consider adding storage via that connection as in a network share. In my particular case, I don't use TM. I use both DAS (directly attached storage a la external 2.5" drives) and also NAS. I find both of these are sufficient for my local needs.
Advice -
If you use a NAS, the biggest challenge most people meet is that NAS drives either are spun down so time is required to spin up and thus get enough out of synch with computers that an operation fails (copy to NAS etc.) or for whatever reason the NAS simply is off line from the computer POV (NAS may be active but the computer cannot see it) and a connection needs to be re-established. There are ways to make sure the NAS is ready before any backup or TM operation.
If you go with external drives directly attached, make sure you have a good cable and note whether the external drive is truly bus powered or should use an external power supply. This can be a failure that runs intermittent when people assume bus power is sufficient and it really is not. Because high speed is really not needed for TM after first time, a decent mechanical drive that is of moderate size might be a good fit for most. Something that is from 1-4 tb depending on real needs.
Drives attached to routers - not my favourite but it does work. I say this as reading from the drive works well but writing to the drives can be rather slow or uneven (meaning if you are copying over both small and large files, the write times for each can be substantially different and take a long time). I tend to find drives on routers great for storing data such as media files or sharing documents between local network computers and users.
We are under the same roof, but we both move our laptops around a lot, so wireless storage would partly mitigate having to worry about dislodging cables or knocking drives onto the floor, even if it meant slightly slower speeds and having to re-establish a connection. I suppose in this respect, an NAS would be the most appropriate option, but it's finding ones that work well with Time Machine that can be the issue. We both already have external drives with backups of important data, like photos and music, but my end goal is to ensure that if one of our SSDs fail for whatever reason, we can get it replaced and immediately recover from the most recent snapshot of the drive to get back to near-enough where we were before. There's probably better options out there software-wise (heard good things about CarbonCopyCloner), but I was hoping to use Time Machine specifically as it's saved my bacon on more than one occasion, and would probably be the easiest for my mother to understand when I'm not around (in our house, if it's technology-related, I get it working/fix it when it breaks).
I have attempted to attach drives to my router (some Technicolor piece of junk that our ISP provided and insists we use 'for the best experience'...pfft), but unfortunately it doesn't want to play nice with HFS+, and doesn't accept HFS+ Journaled formatted drives. The best I can get from it is FAT32. :/
The fallback solution, I suppose, would be to give in get one of the now-discontinued Airport Extreme and attach drives, or just bite the bullet and buy a Time Capsule, but that's quite an expensive option for 3TB when I already have a couple of 2TB external drives lying around. I know going the NAS route won't be much better cost-wise initially, but I had assumed that once I had an appropriately compatible enclosure, I could just swap larger drives in and out of it as required. Whereas with the Time Capsule, I'd be limited without cracking it open and potentially damaging it.
I'm likely confusing both myself and anyone looking at this, so I'm going to shut up now.
TL;DR conclusion: I think what I'm looking for is a Time Machine-compatible NAS?
