When I disassembled the faucet, I found that the outside housing of the ceramic cartridge that controls and mixes the water flow in the faucet was broken in several places, causing the leak.
Removing the cartridge was quick and easy. It involved first removing the little hot/cold indicator button on the handle (prying it out with a small knife blade), revealing the hole to the handle set screw. After loosening the set screw the handle can be removed.
Next, the rounded top collar can be unscrewed (in my case I was fortunately able to do it by hand). If it's stuck I'd use some penetrating oil first, along with a rubber work glove to get a better grip. If you need a wrench, I'd suggest a rubber strap wrench as a pliers will likely gouge and scratch the part.
That reveals the cartridge retainer nut that can be removed with a wrench.
Once I got the cartridge out I found it was marked as a DL02GJ and was dated 06/05/05. It was cracked on both sides, and the top collar ring had separated from the rest of the housing.
I then did what I normally did when looking for parts - an Internet search using the part number. That wasn't much help, only locating Chinese and Taiwanese manufacturers for the part (for instance, Hain-Yo Enterprises Co. makes the cartridge) and dimensions for it.
Since we had bought the faucet at Menards, I thought they might stock the part. But a perusal of their replacement cartridges online (they only sell 3 different models) didn't show anything that resembled the one I had (but then there are few details on their listings as well).
Since at the time I wasn't sure what the manufacturer and/or model number for the faucet was, I had to do more investigatory work online. I first determined it was a "Tuscany Brass" brand, which according to this thread is made by LDR Industries (thanks, guys!)
An examination of the LDR catalog showed it was an LDR 950 10504SS model. Then I searched using the manufacturer's part number for the faucet and other than listings for sale of the faucet itself found no parts for it.
I can thus only conclude it must be considered a disposable faucet, but we're in the process of a remodel and I didn't want to spend the time nor money to replace the faucet right now. So given no other options, I made a run to Menards to see if one of the few ceramic cartridges looked similar to the DL02GJ I had.
Not only are the cartridge outside dimensions important, but the bottom has a gasket and two pins that must match the faucet assembly.
The 931_T300 (Menards sku 6740348) looked very similar; especially the bottom layout and gasket, except it had this red plastic top collar. (FYI - if you do a search on the Menards website for "ceramic cartridge" you won't find this part, because it's listed as a "cermaic (sic) cartridge").
I found however, that the red collar can be removed, and the part ended up fitting perfectly. I quickly reassembled the faucet and the leak was gone.
Here's a photo of the old cartridge, the red collar from the new one and the packaging card for the replacement part.
Of course, I did a follow-up online search on the 931-T300 part number and also drew a blank.
I'm considering buying another spare as a future replacement, given my difficulty in locating the part, but by the time that happens we'll likely be replacing the faucet.
Hopefully this post will save someone some time in locating this replacement part. The 931-T300/931_T300 ceramic cartridge (sold at Menards) is a replacement for a DL02GJ.
Update 1/27/14 - Thanks to Mike, who commented that this part is now available for purchase from Menards online with shipping to home.