Are you running this on your own server? If so, check php.ini for the mysqli extension entry (which is probably commented out with a # in front of it). Uncomment it, and restart apache (just to make sure it sticks)
Do you really need phpmyadmin? It’s quite heavy tool for basic mysql… My favourite is Adminer, it’s only one file to upload and no installation needed. You should try it out…
Regarding your problem, I think the problem is in PhpMyAdmin basic settings. Is there a line about Mysqi? I don’t think it’s required