From this Web site, http://www.word-detective.com/091400.html

they say

Weeds" the plants and "weeds" the mournful getup worn by widows in days of yore are two entirely separate words with unrelated origins. The "ugly, worthless plant" kind of "weed" comes from the Old English word "weod," which meant "grass, herb or weed."

"Weeds" meaning "mourning clothes," on the other hand, comes from a very old Germanic root meaning "clothing," and when this "weed" first appeared in English around A.D. 888, it was used in the singular to mean simply "an article of clothing." By about 1297, "weed" or "weeds" meant a style of clothing typical of an occupation or station in life. One might speak of a priest's "weed" or a beggar's "weeds," for instance. The phrase "widow's weeds," denoting the black veils and other accoutrements of deep mourning, first appeared around 1595, and is the only use of "weeds" in this sense still commonly heard in English.

From this Web site, http://chestofbooks.com/reference/Dictionary-of-Dry-Goods/Widow-s-Weeds.html

they say

Widow's Weeds

Widow's Weeds. "Weeds" signifies a garment, or a dress. The term is now almost obsolete, save in its retention to describe the mourning dress of a widow. The custom for widows to wear a peculiar style of cap is of Roman origin, and the wearing of some sort of weeds was compulsory for ten months. The term "weeds" first began to be used during the middle ages to signify an entire dress.

And if you want, you can go to,
http://en.wikipedia.org/wiki/Mourning

and read more.