The correct option to complete the sentence is lay

In this sentence, we need to use the correct past tense form of the verb "lie" to describe the beggar sitting down by the roadside when he was tired. 

The past tense form of "lie" (meaning to recline or rest) is "lay." Therefore, the sentence should be: "When the beggar was tired, he **lay** down by the roadside."

• "Lied" is the past tense of "lie" when it means to tell an untruth (not applicable in this context).
• "Laid" is the past tense of "lay" when it means to place or put something down (not the intended meaning in this sentence).
• "Layed" is not a correct verb form.
• "Lain" is the past participle of "lie" (not used in the past tense).