This is common when adding other drives such as thumb flash drives or flash cards etc, as well as sometimes Windows on its own decides that one drive comes before the other.
What I would do is shut the comnputer off and unplug these 2 drives. Then boot the system and the drive that you want to be G: plug that one in first, and then the drive that you want to be H or I plug in second, then while the system is running it sticks with the same drive letters.
I find it odd that your drive is skipping H: is there a mapping for H: elsewhere for it not to be G: and H: and instead jumping from G: to I: for both drives?
In the past I had some issues with a CF Card Reader and External Hard Drives moving around on me. To fix this, I told Windows to make the external drives have letters at the opposite side of the alphabet such as Y: and Z: since Windows by default wants to give the next available driver letter in alphabetical order to whatever gets attached next etc. If you are careful you can change the drive letter to be permanently Y: and Z: and this problem should be gone for good.
More info here:
http://smallbusiness.chron.com/change-drive-letters-external-hard-drives-windows-8-67525.htmlBe sure to be careful not to format the drive or delete the partition as for this will destroy your data. So backup your data to another drive if you can before doing this, although most of the time the external drives data is larger than can be back up copied elsewhere and so you sometimes just need to be careful and take the chance on it.