It's tough finding really attractive wines under $10, but it isn't impossible. One of the tricks in finding real values is to look where few others are. This almost makes it impossible to meet some of the gifting criteria I laid out, but if you want to shop at this price point, you're going to have to rely on the quality of the wine to make the gift wow.
I would suggest Portugal as one place to look. Great wines are being made, but the grapes are unfamiliar and the country has a bit of a mixed reputation for wine, so consumers tend to tread lightly. My recent favorite has been the fantastic 2008 Cabriz Colheita Seleccionada Dao.
Though other options can be found by reading 8 Great Wines of Portugal's Alentejo.