The correct answer would be 'that' but actually 'This is the last time I'll give you a lesson' is correct without 'that' because we can omit the conjunction 'that' when it is used to introduce a subordinate clause.

Neither makes sense. This'll be the last time (that) I give you a lesson.

"That" - the reason is that "I give you a lesson" describes "the last time". Imagine if you heard "This is the last time." and nothing else. Your question would be "The last time for what?" Since you asked "what", that means that you need something to identify which last time it is - so you need "that". It's like saying "This is the car that I drive." You are identifying which car it is that you are talking about. You would use "when" if something is happening at the same time. "I'll give you a copy when I give you the lesson." You give the copy at the same time of the lesson.

use that

Final lesson is that the last emphasized, not time, so in the blanks should consider that a little bit, so use "which" is more appropriate.