sich besinnen auf
to recall
To remember something past or forgotten; to bring something back to mind.
Ich musste mich einen Moment besinnen, woher ich diesen Mann kannte.
I had to think for a moment to recall where I knew this man from.

to reflect / to recall
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to recall
To remember something past or forgotten; to bring something back to mind.
Ich musste mich einen Moment besinnen, woher ich diesen Mann kannte.
I had to think for a moment to recall where I knew this man from.

'Besinnen' is always used reflexively, meaning with 'sich' (oneself). The reflexive pronoun must be adapted to the person: ich besinne mich, du besinnst dich, er/sie/es besinnt sich.
Ich muss mich kurz besinnen, bevor ich antworte.
I need to reflect briefly before I answer.
The verb contains the old stem 'sinnen', which means 'to think' or 'to ponder'. The prefix 'be-' intensifies or directs this action.
Er saß am Fenster und schien sich auf etwas Wichtiges zu besinnen.
He sat by the window and seemed to be recalling something important.
In the meaning 'sich erinnern' (to remember), 'besinnen' is always used with the preposition 'auf' and the accusative: sich auf etwas besinnen.
Sie besann sich auf die alten Traditionen ihrer Familie.
She recalled the old traditions of her family.
The idiom 'sich eines Besseren besinnen' (to think better of it) uses the genitive ('eines Besseren'). This is a fixed, unchangeable structure that should be learned as a whole.
Zuerst wollte er die Reise absagen, doch dann besann er sich eines Besseren.
At first he wanted to cancel the trip, but then he thought better of it.
to consider
'überlegen' and 'sich besinnen' both have to do with thinking. 'überlegen' is the normal word when you are looking for a decision, examining something, or making a plan. 'sich besinnen' sounds calmer and somewhat elevated. It often means that you pause, think clearly again, or reconsider an earlier intention. The words can be confused because both can lead to a new decision. In everyday life, 'überlegen' is usually more natural; 'sich besinnen' fits more when someone comes to their senses or consciously changes their attitude.

Ich habe mich in der Pause eines Besseren besonnen und das Training nicht abgebrochen.
During the break, I thought better of it and did not quit the training session.
Am Wochenende überlegen wir, wohin wir mit den Kindern fahren können.
At the weekend, we are thinking about where we can go with the children.