Bus lanes are commonly implemented to ensure absolute priority for buses at signalized intersections. However, while prioritizing buses, existing bus lane management strategies often exacerbate traffic demand imbalances among lanes. To address this issue, this paper proposes a dynamic Multi-Function Lane (MFL) management strategy. The proposed strategy transforms traditional bus lanes into Multi-Function Lanes (MFLs) that permit access to Connected and Automated Vehicles (CAVs). By fully utilizing the idle right-of-way of the MFL, the proposed strategy can achieve traffic efficiency improvement. To evaluate the proposed strategy, some experiments are conducted under various demand levels and CAV penetration rates. The results reveal that the proposed strategy (i) improves the traffic intensity balance degree by up to 52.9 under high demand levels; (ii) reduces delay by up to 80.56% and stops by up to 89.35% with the increase in demand level and CAV penetration rate; (iii) guarantees absolute bus priority under various demand levels and CAV penetration rates. The proposed strategy performs well even when CAV penetration is low. This indicates that the proposed strategy has the potential for real-world application.