What a beautiful pen,
@CarolOMalley -- It looks like it's in pretty good shape, despite its age. It could be cleaned up and put back to use for sure, but it will take a bit of patience and elbow grease. I refurbish a lot of vintage pens.
The only concern I would have is that the nib itself has been worn down so that it is now essentially oblique, but as long as you too are right-handed, you could still be able to write with it - but you'll need to hold it at the same angle your great-grandmother did so that the slit makes contact with the paper. It won't make super fine hairlines anymore, and it may affect the ink flow and flexibility - a nib of this vintage is going to have some flex to it - what fun!
You are correct that this is an eye-dropper fill pen, which is also good news, because it doesn't have a decaying ink sac or a syphon mechanism that might need repairs. And you will be able to use almost any ink that says it is "safe for fountain pens" -- but I personally would stay away from any ink with particulates: pigmented ink, ink with shimmer, and iron gall ink (even if it says its safe).
So here's what I'd do to refurbish it, if it were my pen. I've probably restored 3 or 4 pens of this vintage. This is a long post, but it has detailed instructions.
1. soak the nib-unit. This is actually three parts: the nib, the feed, and the grip that holds the nib onto the feed. For the pen to ever work again, you need to get all the dried ink out of the feed - and that probably means you should try to take them apart. But they'll be hard to get apart. Soak them together in plain water for a day. Then dilute a bit of kitchen ammonia 1-1 and soak them in that for a day or two (do not use alcohol; it will dissolve the grip, which is most likely made of resin). Keep refreshing the bath until the solution stays clear overnight.
2. while the nib-unit is soaking, clean the body and cap -- soak it in some plain water and use a long-stem cotton swab or a pipe cleaner to scrub the insides. Use another cotton swab to dry them out thoroughly. You could polish up the silver-work on the outside with a jewelry cloth too.
3. Do you have an ultrasonic jewelry cleaner? Like this:
https://www.amazon.com/Magnasonic-Professional-Ultrasonic-Eyeglasses-MGUC500/dp/B007Q2M17K/ref=sr_1_8?crid=3ECKZR5MVR1OD&dchild=1&keywords=ultrasonic+cleaner&qid=1623026544&sprefix=ultra%2Caps%2C207&sr=8-8 They're not very expensive, so if you don't have one, but really want to refurbish this pen, I recommend buying one. They are a fantastic way to clean nibs and brushes too, so it's not a single-use item. I use mine almost every day. If/when you have one, put the nib-unit into into a small glass with some of the ammonia solution, and set the glass into the reservoir, and fill the basin with water to the fill-line. Set the cleaner to run on its longest setting. Ink will come out of the unit. You can put it through several more cycles, but let it cool down in between so you don't overheat the motor.
4. Eventually it will look like there's no more ink stuck in there. You'll need to change the solution inside the glass a few times. When it looks like it's as clean as it'll ever be, use a small cloth and gently pull the nib & feed from the grip. Use only your fingers. Do NOT use pliers; you will crush the feed. The feed is the most delicate part of the pen. If it won't come out - do not force it. Go back to soaking/vibrating it. If you cannot ever get it out, can still be okay and useable -- but taking the nib unit apart is the only way to check to see if the nib is corroded underneath, and clean or replace the nib if there is corrosion.
5. Let the parts dry on their own for a day or two if you got them apart. When you put it back together, make sure the feed lines up with the nib (the "shoulders" should align). If you didn't take it apart, you can fill the body with some "fountain pen ink" (see above) and see if it works. You may need to put it in a cup, nib pointed down (with the lid on), to let gravity get the ink into the feed. You may need to adjust the nib & feed relative to each other to adjust the ink flow.
6. Test your pen to make sure that nib works for your hand. If it doesn't, you might be able to find a vintage replacement on eBay or a modern nib that fits.
7. Enjoy your refurbished pen! Note - every fountain pen has inks that it works best with and worst with, so you might order some sample vials to see what works well with yours. Pen shops like Goulet, Anderson, VanNess all sell 2ml to 4ml vials of ink for $1-$3 each.
If all of this sounds too intimidating, you could try contacting a pen shop that does repairs -- like Anderson:
andersonpens.com. Or feel free to message me if you're having difficulties.
Best wishes for restoring your beautiful pen - and please share updates about outcomes!
--yours, K