The emergence of functional oligonucleotides on early Earth required a molecular selection mechanism to screen for specific sequences with prebiotic functions. Cyclic processes such as daily temperature oscillations were ubiquitous in this environment and could trigger oligonucleotide phase separation. Here, we propose sequence selection based on phase separation cycles realized through sedimentation in a system subjected to the feeding of oligonucleotides. Using theory and experiments with DNA, we show sequence-specific enrichment in the sedimented dense phase, in particular of short 22-mer DNA sequences. The underlying mechanism selects for complementarity, as it enriches sequences that tightly interact in the condensed phase through base-pairing. Our mechanism also enables initially weakly biased pools to enhance their sequence bias or to replace the most abundant sequences as the cycles progress. Our findings provide an example of a selection mechanism that may have eased screening for the first auto-catalytic self-replicating oligonucleotides.