Learn French: J'ai travaillé ici pendant mon séjour.

Pendant means during or for when talking about a specific, closed period of time. Because the stay (mon séjour) is a set event with a clear beginning and end, pendant is the exact right word to use.

Could I use pour instead of pendant to mean for my stay?

Not in this context. Pour is used for intended future durations, like saying you are leaving for a week. When talking about a past, completed duration or saying during an event, you must use pendant.

Why use the passé composé (J'ai travaillé) rather than the imparfait (Je travaillais)?

The phrase pendant mon séjour (during my stay) puts a strict time limit on the action. It tells us the work had a clear start and finish. For past actions enclosed in a specific timeframe, you use the passé composé.

Is the word séjour related to the word jour?

Yes! Jour means day, and a séjour is a stay or visit somewhere. Thinking of it as spending days somewhere is a great way to remember it. Because it is a masculine noun, we say mon séjour (my stay).