The cyclic frustration that is my old Nexus 7. As I install more apps and things get updated, the poor thing eventually struggles to keep enough RAM to remain responsive. Consequently, I have tried a few ways of coaxing it into doing things a little faster.
Installing a fresh OS (be it the factory software or otherwise) keeps things smooth for a while; however, the accumulating apps, data, or updates (I'm not sure which is the most problematic here) does tend to slow things down.
Recently I installed Sailfish OS. It seems fast enough, but I managed to kill the OS when installing Japanese input. Whoops.
Anyway, if you care about not voiding your warranty, and have a rooted Nexus 7 try this:
1) Install MultiRom Manager
2) Download the Cyanogenmod and Sailfish OS images linked at this page.
3) Copy the downloaded files onto the home directory of the nexus
4) Run MultiRom Manager and have it install all the necessary kernel patches etc.
5) Once (4) is finished and you're back in Android, use MultiRom Manager to reboot into recovery
6) Go to "Advanced" and find "MultiRom" then "Add ROM".
7a) You can now follow the pictures.
7b) i) If you dislike pictures; select "SailfishOS" as the ROM type and press next
ii) Select the CyanogenMod and SailfishOS zips (in "/data/media/" if I recall correctly)
When booting up, you can interrupt the boot countdown and then select SailfishOS from the list.
EDIT: I did also try "Plasma Active" which feels much more "old-school linux"; however, its small keyboard led to immediate removal.