I've been using org-mode for the last few months to do my time tracking and it has worked out pretty well. You can use "clock tables" to get a report of all of the logged time within a certain window (e.g. february)
Works great, especially if you're using emacs for writing code too.
Be diligent in describing how the time you have spent has resulted in a benefit for your client.
I found that our clients are interested in quarter of day granularity. E.g
Thursday 2nd Feb 2012, 4 hours, Installing and configuring Operations' Team new Wiki website.
Thursday 2nd Feb 2012, 3.5 hours, Fixed problem where the CEO's iPhone wouldn't sign in to the reporting website.
To see what works for you, send your client your best effort on a time sheet for the first two days or so, and get their feedback. Its better than waiting an entire month to find out your reporting process isn't up to scratch.
Sarcasm aside, is this the type of job where you will be working on a number of projects each with their own pot of money that has to be kept separate? Or do they just need a total figure?
Either way, pencil and paper work for me. Heck, I have to keep track of hours by project and I work on-site. Never done anything more complicated than notations on scrap paper for my own use, and the company timecard program at the end of the day.
I recently started using Toggl (https://toggl.com) and have been enjoying it a lot so far. Type in a description of what you are doing, hit start button. The usual features like multiple clients/projects, billable vs. nonbillable, reporting, etc.
Harvest and Freshbooks also get a lot of love. If you just need a really simple timer, SlimTimer is lightweight and easy.
Look here at http://orgmode.org/ and http://doc.norang.ca/org-mode.html for brief and in-depth use information.
Good luck.