To think over something is to consider a decision carefully before answering. Think it over, think over the offer, take a day to think over.
It buys time politely, and asking to think over an offer is never rude. The over adds the sense of turning the thing round and looking at every side.
1phrasethink overTo consider a decision carefully before answering.
“Take the weekend to think over the offer.”
think over is a phrase.
Also accepted in games: mull over
Practised in Phrasal Verbs