s_Measure is one of the IE in the rrcConnectionReconfiguration (atleast R8 onwards) -
(Serving cell RSRP threshold controlling whether or not the UE is required to perform measurements of intra-frequency, inter-frequency and inter-RAT neighbouring cells.)
It tells UE when to start measurement. S-measure can have one value from 0 to 97. 0 signifies measurement is disabled. If S-measure has non-zero it means once UE receives RSRP more than value of S-measure for primary cell, it starts measuring cells as per configuration (Measconfig) given by eNodeB in rrc connection reconfiguration message.
Now coming to Hysteresis and offset (skipping threshold its just a limit)
Hysteresis: There can be a delay between action and reaction of a measurement and handover takes place. So there is a small error included in the measurement reporting to minimise the effect of the delay. In short measured neighbor look worse than measured to make sure it is really stronger before the UE decides to send a measurement report to initiate a handover.
Check the following http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6482433
Offset: Each UE evaluates the Event A3 condition every time a new averaged measurement sample is available. The evaluated condition is the entering condition of Event A3: whether the RSRP measured from a neighbouring cell becomes an offset better than the RSRP measured from the serving cell.
