To be fair I would just tell her, you have to many cousins to make quilts for.

I have only distant cousins I have never met.

Boy that's a tough one. My first thought was that not everyone likes quilts so that may eliminate some, but then again maybe not!
Could you possibly offer to teach her how to make one? You could explain that it would be really hard to make one for all the cousins. (what you told us) Sounds like you have a lot of cousins.

If it were me, I’d do this: she can pick a quilt pattern from a selection you would be comfortable making, then meet her at the quilt store, then advise her on the fabric, batting and backing, thread. Add all costs up before any cutting is done. Will you finish it or have it quilted by a long arm service ? Add in that charge too. Maybe she’ll back out or if she wants to stand the cost, then at least she’ll get something in the color & pattern she wanted.

Not sure how close you are to this cousin - do you go to her house and you know that she truly appreciates having a quilt. Some people gush over things like quilts but you go into their home and it is highly decorated with everything matching. The quilts I make are my taste. Usually it is something that reminds me of the person. I don't want to take orders because it is very hard for me to finish something that I don't like.

I have too many quilts I want to make for recipients of my own choosing to be able to entertain requests from anyone else not on my personal bucket list. Life's too short to spend your leisure/quilting time doing something you don't want to do. When I worked someone else told me what to do, when to do it, and how they wanted it done. On my quilting/free time, I'm the boss, no one else. I just tell anyone who makes such a request that I can put them on my list but it's already so long I'll never reach the end. And that's the truth.

You love making quilts and giving them away. You have a cousin who is begging for a quilt. If I were in this situation, I would make her a quilt and not worry about the other cousins. If they also start asking for quilts, put their names on a list and tell them their names are on the list. Then go down the list at any pace you want. If anyone asks why so-and-so got a quilt first, tell them she was the one who asked for a quilt first. Others will have to wait their turn. There is no obligation to make every cousin a quilt before you die; you just make as many as you want to make. And you give quilts to those you want to give quilts to.

Here is why I feel this way. When I was growing up, my favorite aunts had a wonderful hand-pieced, hand-quilted yellow Dresden fan quilt that I absolutely loved. It had been made by their mother. It seemed so valuable to me, I never even thought of asking for it. My aunts never knew how much that quilt meant to me, so they gave it to a friend who was moving away. Had I spoken up even once during all those years, they would have given the quilt to me.

This is why I respect the cousin who is asking for a quilt. I do not necessarily think she is asking for you to spend money and time on her. If she is like me, she highly values your quilts and it would mean a lot to her to have one from you. The fact that out of all of your many cousins, she is the only one to speak up, may signify the gift would have great meaning for her.

So, my vote is to make a quilt for her. If other cousins start asking for one, so be it.

Several thoughts.

I only make the things I want for whatever reason, to gift, to donate, for my dogs etc.

I am not asking for details, but can you afford to make a quilt for this cousin? Have you added up all the costs and really thought about it?

Is she pushy by nature? Nothing like someone telling me to do something, to get me to dig my heels in.

Would your other cousins even know you made her one? Would she go boasting about it? Do the cousins get together regularly? Me, I only see my cousins every couple years.

I do not let other people make me feel obligated to do their bidding. I do not play the game of you did x for Sue, so you have to do the same for Kim. If I do something for one person, it is because I want to. I do not feel obligated to do the same for anyone else.

I never quite understand this concept. A quilt is a special gift. I give gifts to whomever I want, whenever I want. After all, it is my time, my fabric, my skill, my special gift to give. Why should I feel obligated to gift one just because I gifted to one. I had an aunt who made quilts, for some reason -- unknown to me -- she made me a quilt. Only my aunt who brought it to me and my mother knew about the quilt. I never mentioned it to anyone. That would have made it less special and possibly put my aunt in an awkward position. I love that quilt because it is special to me. Reasons for gifting our time, labor, and fabric, should be our choice -- it should never be dictated by the expectations of others. That lessons the value of a true gift.

I think this is great solution!

You stated “ I love making them and giving them away.” Make her a quilt and give it away. If another cousin asks, make another. Do it because you want to.