I have a one-handle bathroom sink faucet (Moen, original to the house, 12 years old).

In the last few weeks, I noticed the faucet drips, so I put a bucket under it to see the volume. After about 10 hours, I was getting a liter or more of water.

Having no plumbing knowledge (and my father is no longer around to help/provide advice), I decided I would just install a new faucet.

After reading the home improvement workbook my father gave me when I bought my first house, I started by turning off the water supply valves under the sink. Here's where I found something I did not expect
When I turned off the cold water supply valve, some blueish liquid (reminded me of ink) leaked from around the valve handle. Also, when the cold water valve is closed, the faucet no longer drips.

I have since turned the valve on & off several times and each turn results in just a few drops of the blue liquid each time. I turned on the cold water at the faucet and noticed a great deal of blushish water coming through, but it quickly cleared.

I have several questions:
1. Why is there bluish liquid coming out of the valve when it is turned on or off?
2. Is this something I can likely repair myself (fairly good with instructions and tools) or should I call the plumber?

My guess is I do NOT need a new faucet (maybe I want one!)

Thanks in advance for your time and advice,
Nashville, TN