Or you can say something to your aunt like, "I love, the bag you gave me and I have been using it almost every day...I'm wearing it out and need to get a new strap for it..."
Depending on your aunt, anyway. One of my aunts would have handed me a gift she'd had in her closet for ten years and would have no idea where it came from and would be embarassed by the question. Another aunt would have spent way too much money at someplace like Coach and would insist I take it into the shop and have it replaced "immediately!"
So, yeah, depends on your relationship with the aunt in question...Or find a decent leather shop and take it in--they might recognize the brand and be able to help you out anyway.