The phrase "lay eyes" implies an act of seeing or looking at something or someone for the first time. There are various synonyms that can be used to express a similar meaning, such as "set eyes on", "behold", "glimpse", "spot", "discern", and "observe". These words can be used interchangeably depending on the context and the desired level of formality. For instance, "set eyes on" may be more suitable for a formal or literary setting, while "spot" may be more appropriate for a casual or conversational context. The use of synonyms can add variety and richness to one's vocabulary and help to express ideas more precisely.