I want to use MySQL to store session variables. From what I understand this means that on every page request there will be one read and one write to the table.
Which MySQL storage engine is best suited for this task? MyISAM, InnoDB , MariaDB (which I don't see in PHPMyAdmin), Memory, or something else entirely?
"Best" means nothing. You need to express your constraints: do you need consistency? Durability? High-availability? Performance? A combination of all these properties? Can you afford to loose your sessions? Can they fit in memory? Do you need to support concurrent accesses to the same data?
Without more context, I would choose InnoDB which is the most balanced storage engine. It provides correct performance for OLTP applications, ACID transactions, good reliability, and sensible concurrency management. Session variables access will likely be done using primary keys, and this operation is very efficient with InnoDB.
Now if performance is really a constraint, I would rather use a NoSQL engine (i.e. not MySQL). To store session data, Redis usually does a very good job, and is easy enough to integrate and deploy.