Is it cheaper for this person to use you than call a taxi Boona? It's a real cheek for these people to expect you to use your car as a cheap taxi. Not only do you have fuel costs but also wear and tear and the hassle of doing the run. Of course, once you start acting as a cheap taxi you feel obliged to do it all the time and the other person expects that. The moment you say you can't do the trip anymore, you are in the wrong!
I had this a while ago whenever I collected my wife from her social club. There was always someone who had asked for a lift "if you are going my way". Of course, they asked my wife - not me! It became apparent the only reason they asked was because it was cheaper for me to drive them home at my expense rather than them call a taxi. They would just get out at the other end, say "Thanks" and walk off! They had travelled home free at my expense and it had cost them nothing! What a cheek!
One night I put a stop to it. When a woman asked my wife for her usual lift home I made a point in front of everyone of taking out my wallet and slapping £3 on the table. I told her that was the cost of her taxi. She was very embarrassed and protested but I told her in front of everyone that it actually cost me less to give her the money than drive her home using my fuel, wear and tear etc. My wife had no more requests from anyone for a free ride home after that!
Perhaps tell your family member that from now on you will contribute towards the cost of a taxi. She will be embrrassed and refuse, but if you explain it will be cheaper for you to do that as you can no longer afford the cost, she'll probably pay her own way in future. Besides. this person obviously doesn't think much of you otherwise she wouldn't take advantage.