Zijlstra is a West Frisian toponymic or occupational surname meaning "from/of the Zijl", a Middle Dutch name for a type of sluice (Modern Dutch Spuisluis, German Siel).
The suffix "-stra" is derived from old Germanic -sater, meaning sitter or dweller.
The name could thus be referring to a lock keeper.
Variant forms are Zeijlstra, Zeilstra, Zylstra, and Sylstra.
[1] People with this name include: Zeilstra Zylstra