Why does the name Savannah have an H at the end?

It's silent. Lots of names have H's, like Sarah or Hannah. Some don't, like Anna or Jessica. It depends on your preference, Savanna is a word so they probably added the H to make it more "name-like" I guess. I personally think it looks sort of plain without the H. Same with Sarah and Hannah. I like Anna without the H so I guess it's everyone's preferences. : )

This name must come from another word, and it transformed through time and now the H at the end if useless. Back then it must have been another sound.

Don't know if it's linked, but lots of other name have H at the end such as Sarah, Hannah, ...

The same reason Sarah does, it's silent. You can spell it without a H if you want to.

In order to give narrow minded people something to whinge about.