If it is the same symptoms as the jeep in the video, it is the centering ball and the u joints that are bad. Not sure what the mechanic recommended, but we recently replaced all the parts in my sons '98 Wrangler Sport front driveshaft.
I like the greasable spicer u joints with the zerk fitting in the end cap (a lot easier for me to get to near transfer case at without removing strap at driveshaft yoke).
You will need 3 of them (2 for centering ball - 1 for front of driveshaft)
http://www.northerndrivetrain.com/product/SPI-5-1310-1X.html
here is the greaseable centering ball kit and the slinger.
http://www.northerndrivetrain.com/product/SPI-211355X.html
http://www.northerndrivetrain.com/product/SPI-231864.html
Also, while you are at it, given the low cost, I would replace the strap kit as well
http://www.northerndrivetrain.com/product/SPI-2-70-18X.html
All together this would put you at about $140-$150 of top quality parts plus whatever labor the mechanic charges. I would think a good mechanic with the right equipment could knock it out within an hour or two.
The only reason I can see that the drive shaft would have to be repaired or replaced is if there is some damage to the nipple that fits into the centering ball. Even if you dont do it yourself, you might want to become familiar with the parts that need replaced to ensure the local mechanic replaces these and to ensure he uses quality parts. The Spicer/Dana components come well recommended (at least from what I read here at Jeep Forum).
My son and I spent a couple hours proceeding very slowly replacing these on his driveshaft. As long as you have the right tools it and spray it several days in advance with PB Blaster or other penetrating lubricant. Lots of videos out there and on Stu-Offroad about how to do this and the tools that you need.
And, or course, the other option as mentioned above is an aftermarket driveshaft (better than OEM) that you could easily replace yourself andmaybe have no more into it than you would have to pay a mechanic to install the pieces that I mentioned above.