I'm surprised that the factory reset doesn't resolve the issue, it always has for me, turning my slow old phones back into shiny new ones again.

However do be aware that your perception of what is slow changes as you use newer devices which perform better, making older ones feel slower than they did before - much like a shiny new work PC makes your home PC feel slow (or vice-versa) - so that may partly explain it.

Finally, I've just breathed new life into my ageing Nexus 7 (2013) tablet by installing the custom ROM "AICP" on it, which has made a huge difference to its performance & battery life. I recommend giving it a whirl on your SO's S3 - download from http://dwnld.aicp-rom.com/?device=i9300 (though you'll need to read up elsewhere about installing custom ROMs if you've never done it before)

My S3 is also about the same age and is running mostly fine, except for two things: 1- the USB port is damaged and the phone sometimes doesn't charge or charges very slowly; I know it's probably a soldering problem, but I can't be bothered to fix it, and 2- the Facebook app frequently crashes and reboots the phone. I suspect this has more to do with Facebook being awful on Android, since I've seen it happening in other phones as well.

Point is, I won't change my S3 until it crumbles to dust, if I can help it. I don't understand the obsession with buying the latest mobile every 1-2 years.

Personal experience suggests that Android (or perhaps Linux?) slows down as the internal storage fills up.

Also, the GC on older Android devices is a mess. It does not take many apps hanging out in the background before things slow to a crawl.

I had some performance issues, but they got sorted out when swapped the battery for a new one.

A battery? How would that help? Not doubting you, just honest question. I feel like I'm missing something very important about how smartphones work.

Mmm... it also happened to me with my (now defunct) Galaxy S1. I can't explain it, but at some point its battery developed the "bloated, about to explode" look and the phone worked but it was very slow and crashed frequently. I changed the battery and everything was ok. Later it died of unrelated causes.

I guess the processor wasn't being fed properly in regards to electricity, meaning not executing at the speed it was supposed to or other side effects.

It's Google services. It's a huge library that's only growing larger.