Is there any way to specify a suggested filename when using data: URI?

Use the download attribute:

<a download='FileName' href='your_url'>

The download attribute works on Chrome, Firefox, Edge, Opera, desktop Safari 10+, iOS Safari 13+, and not IE11.

Chrome makes this very simple these days:

function saveContent(fileContents, fileName)
{
    var link = document.createElement('a');
    link.download = fileName;
    link.href = 'data:,' + fileContents;
    link.click();
}

HTML only: use the download attribute:

<a download="logo.gif" href="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7">Download transparent png</a>

Javascript only: you can save any data URI with this code:

function saveAs(uri, filename) {
  var link = document.createElement('a');
  if (typeof link.download === 'string') {
    link.href = uri;
    link.download = filename;

    //Firefox requires the link to be in the body
    document.body.appendChild(link);
    
    //simulate click
    link.click();

    //remove the link when done
    document.body.removeChild(link);
  } else {
    window.open(uri);
  }
}

var file = 'data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7'
saveAs(file, 'logo.gif');

Chrome, Firefox, and Edge 13+ will use the specified filename.

IE11, Edge 12, and Safari 9 (which don't support the download attribute) will download the file with their default name or they will simply display it in a new tab, if it's of a supported file type: images, videos, audio files, …