The prototype of a time digitizing system for the BESIII endcap TOP (ETOF) upgrade is introduced in this paper. The ETOF readout electronics has a distributed architecture. Hit signals from the multi-gap resistive plate chamber (MRPC) are signaled as LVDS by front-end electronics (FEE) and are then sent to the back-end time digitizing system via long shield differential twisted pair cables. The ETOF digitizing system consists of two VME crates, each of which contains modules for time digitization, clock, trigger, fast control, etc. The time digitizing module (TDIG) of this prototype can support up to 72 electrical channels for hit information measurement. The fast control (FCTL) module can operate in barrel or endcap mode. The barrel FCTL fans out fast control signals from the trigger system to the endcap FCTLs, merges data from the endcaps and then transfers to the trigger system. Without modifying the barrel TOF (BTOF) structure, this time digitizing architecture benefits from improved ETOF performance without degrading the BTOF performance. Lab experiments show that the time resolution cif this digitizing system can be lower than 20 ps, and the data throughput to the DAQ can be about 92 Mbps. Beam experiments show that the total time resolution can be lower than 45 ps.