I think that you might be approaching this from the wrong angle. I feel that if you want to encourage people to log hours everyday, you're looking for less "Fun" and more "Habitual".
I would take a page from the book of more causal games, which thrive on getting players to log in everyday, and then leave after 5 to 10 mins.
One way of doing is every time someone logs their hours, they open a mystery box. They have a chance of getting something nice instantly, like coffee, or a currency that could build up to nice things. This is a fairly simple variable ratio schedule, which is a sinner box technique. If it strikes your fancy, you can read up on reward schedules here:
http://en.wikipedia.org/wiki/Reinforcement#SchedulesOr you can take the RPG approach, and turn it into a leveling system. The more consistently you log hours, the more XP you get, and every level is a nice thing. For bonus points, you can reward the highest level people in the company with something extra, as a way of setting an example.
There are a lot of ways you can go about doing this, but I would recommend steering away from something that's complex. All you really need is a couple graphics, and a nice "Ding!" noise to encourage most people.