Upgrading just the memory, for example swapping the logic board for another iPad mini2 logic board with more memory, will not make you iPad any faster, it will just hold more stuff.
Replacing the logic board with one from a more recent iPad mini, for example the iPad mini 3, will not make much of a difference because they essentially share the same specs and components (CPU, Graphics etc). The two versions of the logic board look identical but I have never tried this so I couldn't say for certain if it would work. However, like I said, it will not really change anything even if you could do it.
The iPad mini 4 logic board is better spec'd but has a different connector layout so you wouldn't be able to use it in a mini 2 housing & peripherals.
Unfortunately, if you want a faster, more powerful iPad, you will need to either buy a mini 4 or get a full-size iPad like the Air 2 / Pro.