"Weigh" here means "influence your decision". "in my favour" means "to my advantage". "Weighing in my favour" means "(A factor) influencing your decision so that you may choose me".

Ben's answer is correct. Here's a bit of background to understand and remember the meaning of the phrase. The "old school" way to weigh things was with a balance scale. You would put a known weight on one side and the unknown weight on the other side. If the known weight was lower than the unknown weight, that meant the unknown weight was less than the known weight and vice versa. From this came the metaphor of "weighing" the value of something by putting the negatives on one side and the positives on the other. So, for example, one could say "X weighs in your favor but Y weighs against you".