。
She came up with a possible solution.
Literal
She [topic-は] possibility [attributive-のある] solution-method [object-を] hit-upon.
可能性のある is a noun-modifying frame built around the existential ある: 可能性 ('possibility') + の (replacing が in the relative clause) + ある = '(an X) that has possibility.' So 可能性のある解決法 = 'a solution that has the possibility (of working).' This ~のある~ frame is one of Japanese's productive ways to compactly attach an attribute to a noun. The verb 思いつく ('come up with, hit upon, think of') is the verb of choice for sudden flashes of insight — the subjective click of a new idea arriving.