Metric time would basically be the metric alternative of the measurement of the passage of time. Since there currently is no standard for metric time, this is my proposal of what I think it should be.
{Note: you should be familiar with the way the metric system works}

What should designate one unit of time?

In going with the tradition of basing metric measurement with other metric measurements through some common element of nature (such as 1 Liter was defined as a 10 cm cube of water), I propose basing the measurement of time off of the speed of light. To place the duration of a single time unit within a reasonably understandable quantity, I arbitrarily decided that it would work best to define one metric time unit as the amount of time it takes light to travel one billion meters (1 x 109 m). For more precision, I am taking the speed of light as the amount of time it takes light to travel in a complete void (e.g. complete vacuum and without gravitational distortion) assuming that is possible to accurately measure.

(Or see di5tortion's writeup below where he puts more thought into it and chooses to use 1 x 1012 m rather than 1 x 109 m. I agree that that might be a better choice for practicality.)

What should the basic metric time unit be called?

I must admit, I am not very good at coming up with names for things, however, the name that made the most sense to me was chron. ('chron' as in the prefix for time, so a chronometer would actually measure time in chrons.) The word chron is only one syllable and easy to pronounce (“KRon”). It also works well with all of the metric suffixes (millichron, megachon, centichon, etc.). As for the symbol of chron in shorthand, I could only come up with the lowercase letter ‘q’ from the English alphabet since it’s not used much elsewhere.

How is a chron compared to the current standard measurements of time?

Assuming the speed of light is something around 299,792,458 m/s, one chron (1 q) would be about 3.33564 seconds.

1 q = 3.33564 sec
1 sec = 0.299792 q

Some common numbers:

One hour = 1079.25 q = 1.08 kq
Time it takes the Earth to make one full revolution (1 day) = 259021 q = 259 kq
Time it takes the Earth to make one full orbit (1 year) = 9460730 q = 9.46 Mq
The speed of light = 1 Gm / q (by definition)

Should everybody switch over to this new system?

Right now it doesn't really make much sense to switch away from our current system. Our current measurements of time is based on the natural astronomic system of Earth, and as long as the majority of the human species dwells on or in the orbit of Earth, it would be best to commonly use the current time system. Metric time would be more advantageous in hypothetical physics problems and deep space travel.