The phrase "lays claim to" means to assert one's right or ownership over something. However, there are many synonyms that can be used to convey a similar message. Some options include "claims," "asserts," "purports," "alleges," "proclaims," and "declares ownership." Each of these synonyms may be appropriate depending on the context and tone of the message being conveyed. For example, "claims" may be a more neutral way of asserting a right, while "declares ownership" may be more forceful and confident. Overall, choosing the right synonym can help ensure that the message is clear and effective.