The aim of this study was to investigate the effect of blackberry and raspberry fruits (1 and 4%) and leaves (0.5 and 1%) on the biological activities of rape honey. Honey and plant material extracts were analyzed regarding total phenolic, flavonoid, anthocyanin contents, HPTLC and HPLC polyphenol profiles, as well as antioxidant activity. The antiviral potential was analyzed against bacteriophage phi 6—a coronavirus surrogate—whereas antimicrobial was tested against S. aureus and E. coli. Blackberry extracts were more abundant in antioxidants than raspberry extracts, with better properties found for leaves than fruits and for cultivated rather than commercial plants. The addition of both Rubus plant additives significantly increased the antioxidant potential of honey by four-fold (for 4% fruits additive) to five-fold (for 1% of leaves). Honey with the addition of fruits possessed higher antiviral potential compared with raw rape honey (the highest for 4% of raspberry fruit and 1% of blackberry leaf additive). Honey enriched with Rubus materials showed higher antibacterial potential against S. aureus than rape honey and effectively inhibited S. aureus biofilm formation. To summarize, honey enriched with Rubus fruit or leaves are characterized by increased pro-health value and can be recommended as a novel functional food.