Is there a way to download a list of all the tweets that mention my Twitter handle?

Twitter has an API that gives you a lot of control if you’re looking for something custom.

I don’t know of an existing tool that’s for your problem specifically.

Twitter’s API only let you search about seven days back, so there won’t really be a tool using that. If you use a desktop client perhaps it has saved all your tweets, or you could look at a browser plugin that does this for you.

Perhaps you could try installing the iMacros browser extension and see if you can do something with that – perhaps you can set it up to keep clicking the “More tweets” button on your profile and copying all previous tweets into a different tab (e.g. paste2.org) to collect them.

You can use The Archivist site. You set it up and leave it to run in the background. I haven’t used it myself but if you just want to search for tweets mentioning your handle, it should work.

Or you can do a search once a week for mentions and copy them into a document. That is probably the easiest way to manage your situation.

There is also open source software you can set up but that would probably be a lot more trouble than you need to go to.