Lucky for me, Andrew Gelman decided to discuss this topic on his blog last week! There I found the following books recommended in the comments:
Applied Survey Data Analysis by Heeringa, West & Burglund
Sampling: Design and Analysis by Sharon Lohr
Survey Methodology by Groves, et. al.
Struggles with Survey Weighting and Regression Modeling by Andrew Gelman
Comments from lots of people and Rejoinder from Gelman