There are several synonyms you can use instead of the term "offshore rig." For example, you could use the words "oil rig" or "drilling rig" to convey the same meaning. Another option might be to describe the rig as a "platform" or "installation" since these terms are also commonly used in the oil and gas industry. Additionally, you could use phrases like "offshore drilling station," "deep sea drilling rig," or "marine rig" to describe the same type of structure. No matter which synonym you choose, it's important to make sure that your readers or listeners understand the context and meaning of the term you use.