I haven't spent nearly as much time in the pyrenees as I would like to. I'd avoid Andorra, I'd also avoid Lourdes unless you are a fan of Catholic Kitsch (I was horrified that they'd destroyed a wall from the home of Bernadette in order to force tourists/pilgrims to walk through a gift shop - I was there with my Spanish grandmother's church group, the only other person who seemed to mind was the other non-catholic in the group).

Anyway, back on track.

One place I adored was the National Park of Ordessa.
http://www.ordesa.net/gallery/index.htm (it's in Spanish/Castellano - but the photos should speak for themselves)

For accomodation, even if it's just the one night, I would definitely recommend one of the area's Paradores (there are about 5 in the pyrenees)
http://www.parador.es/english - click on the map and then on the icon for the parador within the map.

You should avoid the winter and the height of the summer, but I am relectant to give further advice as this isn't a part of Spain I know very well and on occasions some parts of the pyrenees have potentially lethal storms and flash floods.

During the summer, many thousands of people walk through the pyrenees on one of Europe's most notable pilgramages - el Camino de Santiago http://en.wikipedia.org/wiki/Way_of_St._James
Even if you aren't interested in hiking this route you should be aware of the dates as accommodation will be very difficult to find.