I found a couple quotes from Steve Jobs and Joel Spolsky that I really liked, so I put them in my Gmail signature. I didn't want to forget them as I found other quotes, so I just added them in as well. However, now my signature is longer than the actual emails I was sending, so I, like the question asker, wanted a way to be able to collect my quotes and inject them randomly into my signature.
I couldn't find a working solution, so I held a little mini, solo hackathon in my apartment that started at about 10pm and ended at 3am, once I pushed the code to Github. You can download Random Gmail Quotes from my Github page and install it as an unpacked extension. To replace the quotes I picked, simply edit the quotes.js file -- in your user profile directory -- and replace them with your own. DISCLAIMER: I wrote the extension.
There is another similar Chrome extension on the Chrome Web Store called Random Quotes, but it's two years old and stopped working. However, unlike mine, it has a nice UI where you can paste in your quotes. So, I installed it, figured out what the problem was, fixed it, and submitted a patch to the author's Github account. If you install this one, and you know what you're doing, you can swap out the bsearch.js file with the one I submitted in the issue ticket I created on that author's Github account.
Hope this helps!
Can't be done as things currently stand. You can't put HTML in your Gmail signature. Frankly, I think this is a good thing. I wouldn't want someone dropping IFRAMES or SCRIPT tags in e-mail to me.
Someone may be able to come up with a way to dynamically create an image from this, but it won't have links and may not work well all of the time.
So, unless and until Google implements this I'd say it's not currently possible.
Best Answer
As of now (03/31/2021), you can set this natively in the web Gmail. I use the signature for "New Emails" and none for "Replies & Forwards."