This lady's passion is all things Dwemer. Books, artifacts, plans, schematics; you name it, she wants it. If you have not already talked to Trebonius in Vivec and got the "Disappearance of the Dwarves" quest, do it. The things you'll pick up in the process of doing Edwinna's quests will help you complete that one.
A simple "fetch and carry" quest. Go find a copy of the book and here's 250 gold to cover it. Some booksellers have copies in stock and you'll undoubtedly stumble across copies in various dungeons and lairs throughout the game. One sure-fire method is to drop by Jobasha's Rare Books in Vivec. He has a copy that you can buy, but any old copy will do.
Bring it back to Edwinna for your 5-point faction rep boost and you can keep the change.
Edwinna has ordered a Potion of Detect Creatures from Skink-in-Tree's-Shade at the Sadrith Mora guildhall. Just use the Guild Guide to teleport there, collect the potion, and return. Skink will tell you that he has a few quests for you, if you're of a mind. Edwinna will give you a Chameleon Potion in addition to the usual 5-point faction rep boost.
Sirilonwe in the Vivec Mages Guild has a copy of a book that Edwinna wants. Your job is to "borrow" it for her. She'll give you a couple of scrolls to help with locks and urge you not to get caught (very helpful advice). Use the Guild Guide to teleport into the guildhall in Vivec and head off to Sirilonwe's room. The book is in a locked chest in her closet. The secret to not getting caught is in doing two things: close the closet door before you try to open the chest and do not talk to Sirilonwe while the book is in your inventory. Read the book for a boost to your Heavy Armor skill (the other book in the chest will boost your Alteration skill) and deliver it to Edwinna. Your reward is the usual 5-point faction rep boost and you can keep the scrolls that you didn't use.
If you need the Mark and Recall spells, you can purchase them from Sirilonwe (don't do it if you have the book in your inventory).
There is some sort of disturbance at Huleen's Hut in Maar Gan. Go find out what's going on and fix it. Take the Silt Strider to Maar Gan (or walk, if that's your preference). Huleen's Hut is visible from the Silt Strider platform (it's just outside the town walls). As soon as you walk in the door, you'll be attacked by a Scamp. This one has a few more health points than a regular Scamp and has a bit wider damage range (1-20 versus 5-15 for a normal one), but isn't too tough. After disposing of the Scamp, you'll notice that Huleen's Hut is a real mess. Search around downstairs and you'll find Huleen's apprentice locked in the storeroom (the Scamp has a key, there is a key on the pile of clothes near the door or you can pick/spell the lock). Seems that he tried a summoning that kind of backfired on him. Anyway, he's grateful for your releasing him and swears he will never summon anything again. Report back to Edwinna for your reward: two 5th Barrier scrolls and the usual 5-point faction rep boost.
Chimarvamidium turned out be useless, so you need to go put it back. The same rules apply: close the closet door before you open the chest and don't talk to Sirilonwe while you have the book in your inventory. Your reward is actually pretty decent this time. One Amulet of Almsivi Intervetion (teleports you to the nearest Tribunal Temple) and one Amulet of Divine Intervention (teleports you to the nearest Imperial Cult Shrine), plus the usual 5-point faction rep boost.
Time for a little dungeon-delving. Edwinna wants a Dwemer Tube and believes you can find one in the Arkngthunch-Sturdumz ruins. If you can handle the 10-point disposition drop from telling her that you don't want to go retrieve it, you can find out about the Chair of Dwemer Studies. I'd recommend saying "yes" and heading off.
The ruins are kind of northwest of Gnisis and west of Ald Velothi. The fact of the matter is that any old Dwemer Tube will do. There is one in the ruins, but there is also one in just about every other set of Dwemer ruins on the map (you might have picked up one in the Arkngthand ruins as part of the Dwemer Puzzle Box quest), plus a few just laying around loose. But there is one very good reason for going to these ruins: the Amulet of Shadows (which is not in the ruins).
Take the Silt Strider to Gnisis and take the road north, following the signs to Ald Velothi. Eventually you will pass underneath of a bridge and come to a three-way intersection. Off to your left you will see a Dunmer woman standing in front of a pond-looking thing. If you talk to her, she will ask you to help her retrieve a ring that dropped into the pond. Go ahead and tell her that you will help her, but don't go into the pond; it's a trap and you'll be ambushed when you find the ring. Instead, follow the road up a little ways to the entrance to the ruins and look down and to your left at the stand of trees. If you look very closely, you'll see someone with a very good Chameleon spell effect lurking near the trees. Use whatever you have avaialble to take this guy down and get his amulet. Both the guy and his girlfriend will attack you, but she's not going to be too much help because she's off down the road a ways. After disposing of these two, you can go look in the pond for the ring (an Expensive Ring). It's near the base of the first set of weeds, but it's difficult to see.
Aside from a couple of critters outside the ruins, there isn't much inside. You'll probably encounter a few Centurion Spiders (all of the critters are leveled, so there might be only a couple), but the ruins aren't that big and the Dwemer Tube is near the back. Not much else in the way of loot in here, though. You get no reward from Edwinna other than the usual 5-point faction rep boost.
More ruins. You need to get an excavation report from Senilias Cadiusus in Nchuleftingth (am I supposed to say "gesundheit" after someone says that?). It's long overdue. Edwinna will mark the ruins on your map, but her directions leave something to be desired. I prefer to start from Molag Mar and just head cross-country since the destination is already marked. If you are going to do that, take levitate spells/potions or Jump spells/potions and be prepared to deal with a lot of Cliff Racers.
Once you get into the ruins, you'll find Senilias Cadiusus and his daughter behind a door to the right of the entrance. No, they don't have the excavation report. It seems that their native guide, Anes Vendu, has disappeared and he had the report with him. Would you be so kind as to see if you can find him? So go explore the ruins. Eventually you will reach a section called the "Test of Pattern". This is supposed to be one of those "one of these is not like the others" kind of puzzles, but I can't see any differences. As you are facing the three cranks, the two on the left will damage you (5-7 points of poison damage for 5 seconds) and the one on the right will open a passage that will let you get to the next section (the Lower Levels).
In the Lower Levels you'll find the body of Anes Vendu, who has the excavation report, as well as a couple or three Dwemer Centurions (Spiders or Spheres, depending on your level). On the floor next to Vendu is a book called "Hanging Gardens" that you will need to complete Trebonius' "Disappearance of the Dwarves" quest. Go back up and talk to Senilias and then report back to Edwinna for your reward: a 10-point faction rep boost and a 1-point overall rep boost.
Yet more ruins and you're looking for plans for a Dwemer Scarab (Redguard players should recognize this). The Mzuleft ruins are almost directly south of Dagon Fel. The easiest way to get there is to either go to Sadrith Mora and take a boat or take the Silt Strider from Ald'ruhn to Khuul and take a boat. You can go cross-country if you wish, though.
The Mzuleft ruins are simply crawling with Orcs, so expect some fairly heavy-duty combat. People looking to get that complete set of Orcish armor should have their wishes satisfied after this one. There is only one section to the ruins, so you shouldn't have too many problems navigating (aside from the Orcs). In addition to the plans, you will also find a book called "The Egg of Time," which you will need to complete Trebonius' quest.
Report back to Edwinna for a 10-point faction rep boost and your next assignment.
The eggminers at the Gnisis Egg Mine apparently broke through into the ruins of Bethamez and there are more blueprints to be had. The Gnisis Egg Mine, though, is closed due to the Blight. You can try to sneak past the guard at the front gate and pick the lock, or (my preference) use the back door. Follow the river around to the left and look for an underwater entrance (called, strangely enough, "Underground Stream"). It's a bit of a way upstream, but not too difficult to find. Once inside, you'll have to deal with a couple of slaughterfish and a couple of leveled Kwama critters. You'll also see an Ancestor Ghost who is tied in with another quest, but he will not bother you. You'll need a Water Breathing spell/potion or two to get in and getting out of the water can be a bit of a struggle, but it's much better than the legal consequences of having to fight guards. You'll also find a book in here that will boost your Athletics skill.
If you choose to use the front-door approach, you need to work your way through the mine to the Lower Eggmine. A good sneak skill (or the Amulet of Shadows) is necessary to get past the Orc guard who will tell you that you have about 10 seconds to get out of the area (and he means it) if he sees you. You can try to run until you get to the Bethamez entrance, but it's difficult and there are a few Blighted Kwama Warriors running around that might slow you down. If you completed the Widow's Land quest for the Legion, you will have already killed this guy.
There are two exits from the Underground Stream. One of them will put you into the Lower Eggmine with the orc directly in front of you. The other will put you into the Lower Eggmine with the Bethamez entrance in front of you, a short way down the passage. When you go through the door and see an orc, you chose the wrong one. In this case, just go back through the door and use the other one.
Once inside of Bethamez, you'll find the airship plans (Redguard players should remember) and a book called "Divine Metaphysics Adapted to the Meanest of Intellects" on a table near the entrance. The book is the third piece needed to complete Trebonius' quest. Aside from a couple of rats, there isn't much else in the ruins, but feel free to explore.
Report back to Edwinna for a 10-point faction rep boost and a 1-point overall rep boost.
If you find a book called, "Secrets of Dwemer Animunculi" (it gives you a spell to summon a Centurion Sphere and is required as part of your Telvanni stronghold), before you turn the book over to the Telvanni builder or sell it off to a bookseller, stop by and talk to Edwinna. She should want to take a look at the book. Letting her do so will get you a 20-point faction rep boost and she will love you forever (+25 points to her disposition).