Van halven bedde literally means “from half a bed.” It is a term indicating half-siblings or half-aunts and -uncles.
For example, if somebody refers to a woman as her “zuster van halven bedde” [sister from half a bed] this refers to a half-sister and a “oom van halven bedde” [uncle from half a bed] refers to a half-uncle.

Hindeloopen interior showing a bedstead and crib, 1892. Credits: Christoffel Bisschop, collection Zuiderzeemuseum, available via Rijksdienst Cultureel Erfgoed (CC-BY-SA)

Example text: Lier, Belgium, Death certificate 216 in 1823:
… zijn verschenen Petrus Van Dessel, bakker, …, broeder van den halven bedde en … van Karel Jacob Joseph Van Dessel, …
Hence, Petrus, who comes to declare the death, is a half-brother of the deceased Karel Jacob Joseph.
Thank you for this post! Today is the first time I came across this phrase in a marriage registration (“broeder den halve bedde de bruidegoms”). Of course, Google Translate and Google were of no help, but this was the first hit when I switched to Google.nl. Of course, I figured it had to do with half-siblings, but I wasn’t sure about the spelling or meaning of the phrase.