The full text of the verse is:
Ein kleines Edelweiß,
das bringst du mir dann mit.
Wenn du von deinen Bergen
Wieder heimwärts ziehst
Ich weiß es ganz genau
Du pflückst es nur für mich
Ich weiß es ganz genau,
dass du mich so sehr liebst
I would not have been able to answer without the full context.
The first issue I am going to address is mit: it cannot be left out as the verb here is mitbringen, not bringen. Bringen is essentially the action of bringing something more or less directly to a person, while mitbringen places more emphasis on the fact that you are doing the route anyway but will bring something back for them.
The second issue is dann: it is, in my opinion, possible to leave it out but having it in makes the phrase sound better. At face value, dann establishes a temporal relation between the time of singing and the time of bringing the singer the flower; a relation that is already established by wenn du von deinen Bergen wieder heimwärts ziehst. That phrase must relate to a future point in time, otherwise it makes no sense and the wieder would be out of place. This renders dann superfluous on the surface.
Nonetheless, it isn’t be wrong to include it; it immediately points out that the singer is singing about the future and you don’t have to wait for the next stanza to find out. In addition and due to the nature of singing, it is possibly not immediately clear where the full stop is and whether wenn du von deinen Bergen wieder heimwärts ziehst belongs to the Edelweiß sentence or whatever is following it. The dann serves as extra reassurance that these two phrases refer to the same (or at least a similar) point in time.
Naturally, omitting the dann would wreck havoc to the iambic meter, so as we have established that mit is necessary there has to be something to fill the gap.