“Sat down at his right hand”
The king stood up to greet her, bowed to her, and sat down on his throne. Then the king had a throne brought for his mother, who sat down at his right hand. (1 Kings 2:19 BSB)
What it sounds like today
Solomon simply being polite and respectful to his mom.
The cultural key
This reads today like ordinary courtesy toward one's mother. It is actually a portrait of an office. In Judah's monarchy the king's mother, the gebirah, held a formal and public rank, which is why the Kings narratives so often name each king's mother and why a bad queen mother like Maacah could be deposed from her position (1 Kings 15:13). Solomon rising, bowing, and having a second throne set for Bathsheba 'at his right hand' stages her authority for the whole court: the right hand is the seat of shared power (Psalm 110:1). That standing is exactly why Adonijah approaches her, rather than the king directly, to press his dangerous request; he is seeking a royal intercessor. The scene is not about manners but about a recognized position of influence beside the throne.
Recovered meaning
A throne at the king's right hand marks the queen mother's formal office (the gebirah), a recognized position of honor and influence, not mere politeness. It is why petitioners approached her to intercede with the king.