Olive leaves as a main byproduct of olive oil and fruit industry are a valuable source of phytochemicals such as polyphenols, with multiple biomedical effects. Apart from leaves, olive branches and stems make up a significant amount of olive waste. It is well known that the drying process and long-term storage affect the stability and concentration of polyphenols present in raw materials. For that matter, two different means of storing olive waste, at room temperature and +4 °C, were compared by determining the content of the polyphenol oleuropein (OLE) in olive leaf, branch, and stem extracts (LE, BE, and SE) by HPLC-DAD method. Total phenols (TPC), o-diphenols (o-DPC), and total flavonoids (TFC) content in extracts were assessed by UV-Vis measurements. LE prepared from leaves stored at +4 °C had the highest OLE content, 30.7 mg g−1 of dry extract (DE). SE from stems stored at +4 °C was the richest in TPC and TFC (193 mg GAE/g DE and 82.9 mg CE/g DE, respectively), due to the higher purity of the extract. The biological activity of extracts was determined on cervical cancer (HeLa), melanoma (A375), metastatic melanoma (A375M) tumor cell lines, and on spontaneously immortalized cell line of keratinocytes (HaCaT), using the MTT assay. The data show that all extracts had a similar dose-dependent effect on cell viability in HeLa cells, while the effect of LE on melanoma A375 and A375M, and HaCaT cells was cell-line dependent.