Hired as a research assistant and told to track working hours for payment, is this normal?
From my experience doing research assistant work at a state university in the US, and from speaking with peers in both "hard" and "human" sciences this is par for the course. A tiny bit of paperwork (a signature or two), and away you go. Though I'm assuming by "she will be paying me" it will be through the university rather than out of her own pocket. 

Do the work (well), do not understate your hours, and enjoy the paycheck when it arrives. I can almost guarantee the researcher is concerned about getting things accomplished in a timely manner, and that the cost of your time is minimal compared to theirs and the other costs associated with the research.