Sorry for the length ...found on one of the links to this page ...I think it's relevant ...
None of my sources specifically mention the derivation of this name, but I looked in an extensive 8-volume Polish-language dictionary I have, which was recommended to me by Polish scholars as a good source of information on terms that often became surnames. It mentions a noun pus~lednik as a different way of spelling po~l~s~lednik (accent over the O, slash through the L, accent over the S), which would be pronounced almost exactly the same way. That noun means "a farmer or peasant who works a 'half' farm." In Polish po~l~ means "half," so this is a "half-farmer."
What that means needs a little explanation. Originally Polish peasants were allowed by nobles to work land that belonged to the nobles. A full-sized farm was one that was big enough to supply food for a family for a year. The size varied from place to place, but that's what a "full farm" was. However, as time went on and property was split among descendants, what began as a full-sized farm might become two half-farms, or 4 quarter-farms, and so on. A po~l~s~lednik was a peasant who owned or worked a "half-farm" -- not one quite big enough to support a family by itself, but still much more land than many peasants had.
Now Pos~lednik (a more common name, borne by 500 Poles as of 1990, with the largest number by far, 267, in Leszno province) probably comes from a different word, pos~lednik, meaning "one who comes after; descendant." However, it can sometimes also mean the same thing as Pus~lednik. Probably the only way to find out for sure which meaning is relevant in a given family's case is through detailed research into their history, which might turn up some information that would shed light on this question.
There are other words in Polish that mean much the same thing, such as po~l~kmiec~ and po~l~rolnik. The fact that Pus~lednik and Pos~lednik are most common near Leszno province makes me wonder if it was a tendency for people in that area to prefer these terms, instead of the others? I don't know, but it does seem likely, in view of the fact that the surnames Pos~lednik and Pus~lednik are most common in that area. This might be a good indication that your family is likely to have come from that area originally.