There are a lot of factors in play as you decide which river to fish for only 3 hours.
If you've never fished the Firehole, then that's where I would fish. It might get a bit too warm, but this summer has been very wet and cool, so you might be ok. I love parking at the Fountain Flats parking lot and hiking upstream a mile or so. There are a few little hot pots and mini geyers that spill over into the water, it's beautiful to fish in that type of setting.
If you want to catch a ton of little brooke trout, then go fish the upper Firehole or the Little Firehole river. However, it appears that there might be road closure / construction issues.
Here's another link that shows a good map of the park: http://d1njyp8tsu122i.cloudfront.net...ark_Map-11.pdf
The drive time from your home base to the desired river(s) needs to be considered. If you can make it work, it's a lot of fun to go fish the Lamar, or Soda Butte in northeast section of the park.
Also, you should go to the Ynet forum and ask the expert yellowstone fans about the construction issues. Check out this site: YNet Forums • Index page
For good updates on fishing reports check this site:
Fly Fishing In Yellowstone National Park
Have fun! Try and work it out to get more than just three hours
Also, go talk to Blue Ribbon Fly shop in W. Yellowstone for more info and to purchase your fishing license. You can't buy them online.