I just did this!
Big help from this thread
You will need to download 1.6 PrestaShop version
Unzip file on your computer
Open up FTP and Navigate to public_html -> autoupgrade -> latest
Upload the 1.6 PrestaShop to that directory
Once completed open up the 1.6 folder and move the PrestaShop folder within it to the latest folder (the folder one above the 1.6 so the 1.6 and PrestaShop are both now in the latest folder
Once done select Expert Mode under the 1-click module version selection then change it to Local Directory and enter the 1.6 version
Run the update
This worked for me and I was able to update to 1.5 then to 1.6.
I used the 1-click module to update from 1.6 to 1.7, but be sure your server to up to date with the system requirements for whichever PrestaShop version you are looking to use. \
I would also advise you to delete/disable any old modules that aren't compatible with 1.7 BEFORE you update, as it is causing a headache for me now.