Sometimes a simple "Save As" will reduce the file size slightly, but probably not enough to pass the <8 mb threshold of your recipient, as you need.
You've probably seen it, try the next option in the Save As menu, "Reduced Size PDF," and make it compatible with the latest version possible, again considering the recipient. This suggestion assumes Acrobat X. You don't say which version you're using, but this tool was available at least going back to Version 8, but in a different place, you'll have to hunt.
If all that doesn't work, you could remove some images from the source document, which also will reduce the overall size.
I've found that when you go to Optimize PDF and check fonts almost all PDF have multiple Copies of the same Font and same font style.
I've worked PDF that say would have up to 5.6.7 copies of say Helvetica Bold Italic for example.
You only need one copy of each font type period. The fewer multiples there are the smaller the PDF will be.
I found your comment interesting because I've looked at tons of PDF files, almost all of them created by InDesign or Illustrator. I can't remember seeing such font duplication.
What applications do the PDF files you have observed come from? Clearly it's a function of the application and how they're creating the PDF file. Adobe InDesign and Illustrator use, I believe, an Adobe PDF Library to create the PDF and they must do it in such a way that font duplications are eliminated. Also, almost all fonts from Adobe applications are subsetted.
Might this multiple font list be a result of the way that MS tends to do things? They HTML that they produced from WORD files tended to be badly bloated, so if the PDFs were produced by the MS plugin for WORD, that might be the issue. I am not claiming that is indeed the case, but might be a possibility. Another possibility might be that the multiple fonts could come from a merging of multiple PDFs. There is also the chance that the fonts are actually different, but have the same name. So there are multiple ways this result could have occurred. Interesting and determining how it happened would be of interest also. What produced the PDFs with the multiple fonts (should be listed on the general tab of the document properties).
I don't own Indesign and although I have Illustrator because I have web Premium 5.5.
I do have and use Acrobat X though. and all I've converted using the save as Adobe PDF (mostly word or Excel Documents but not all, some others), end up with duplicate fonts. Often removing duplicates will drop the size by as much as half. I've seen this also in FileMaker Pro. And, OpenOffice its variations.
Another thing that will help is reducing dpi of any graphics. to a 150 dpi