[Pinax depicting a funeral, 6th cent. BCE, photograped by the Walters Art Museum]
[Source: Wikimedia Commons]
If a widow didn’t remarry, it was generally because she was past childbearing age. This was a precarious position to be in. A widow with adult sons who did not remarry would usually have one of them for her kyrios, or guardian. The law provided for protection from abuse by her sons. How reliable that protection might have been seems likely to have depended on social class at the very least. Childless widows and those with underage children were dependent on the power of their male relatives and only those with children were permitted to stay in their late husband’s household. None of this is to say that they were completely powerless, but they were dependent on the standing they had with those relatives.*
Outside of Athens it seems that widows had more control over their own lives. At the very least, many city-states allowed women to own property in their own right, even if they couldn’t dispose of it without the permission of their kyrios.** In the city of Gortyn a man could not leave property to anyone but the legal (male) heir except to leave a gift to his wife. City-states other than Athens also seem to have had less of an emphasis on remarrying a widow right away, probably in part because they at least allowed women to own property.
*A good example of this is the case of the widow of Diodotus, who with the help of her other male relatives, took her kyrios to the task for recklessly spending her underage sons’ inheritance. We also see widows working to support themselves or their children. Nearly all of the women we see making business transactions were non-citizens, childless widows and widows with young children.
**This also seems to have meant that a woman’s kyrios couldn’t dispose of it without her permission.
-Mary R. Lefkowitz and Maureen B. Fant, Women's Life in Greece & Rome: a Sourcebook in Translation, 1992. (The index is unfortunately lacking the term "widow" for some reason, but you can search the book for it here on Google Books. Most of the sources in there refer to Rome, but there are a few from Greece.)
-David M. Schaps, Economic Rights of Women in Ancient Greece, 1979.