Here are some options to consider.
- Software for managing downloads such as the ones reviewed here: Download managers Look for something like JDownloader which has a field to leave a comment about what you've downloaded.
- A spreadsheet or table. Make a field for the filename, a simple description, notes about why you wanted to download it. Some genealogists like Thomas MacEntee are big fans of 'only touching a record once' and have elaborate logs -- but it's important to remember that the best logging system is the one you'll actually use.
- Dedicated source-centric genealogy software such as Clooz or Custodian.
- A document in your favorite text editor, word-processing software, or Scrivener. I especially like Scrivener because of its integrated outline, editor, and index card views.
The trick with any logging workflow is establishing the habit of logging all the bits and pieces as they come in. For my genealogy webinar handouts, I keep a database of the webinars I'm interested in watching, so once I do have the link to the handout, it's simple to add the link I downloaded it from, and to link to the folder I stored it in. Most of the work has been done in advance. But your problem is different because these aren't items that you know about in advance. So my advice would be, make a simple log where you can easily drop in the name of the downloaded file and a brief note about why you wanted to keep it. Don't get too elaborate -- keep the log simple so you'll use it.
P.S. You referred to files which are not properly named. Unless the file downloads with some generic filename like "Document.pdf" which has to be renamed, I don't rename my genealogy files. Several of the sites I use put the archive reference in the filename, and keeping the original filename helps me see whether I already have a copy of the file. Use what works for you.
Edited to add: special caution if you decide to use a spreadsheet: I highly recommend only sorting a copy of your log, not the original. If you don't have the entire area of your data selected when you do the sort, you can break the association between the cells that belong to each row. If you save the badly-sorted results without noticing what you've done, you've destroyed your log -- your only hope of recovery is to go to a backup. Sorting a copy of your worksheet instead of the main log protects against those "oops, not what I meant to do" moments.