"Fire-Hair" might be more accurate. She's the Guild Steward for Balmora and you'll find her upstairs. Once you have reached the rank of Defender, she is available for training in Blunt Weapon and Axe (72%) and Long Blade (67%).
Eydis is a supporter of Sjorring Hard-Heart, the Guildmaster in Vivec, so her quests will seem slightly off from the norm, but they start simply enough.
Other training and services are available in the Balmora Guildhall:
This quest can kick off one of two ways. You can either pick up the quest from Eydis or from the client (Drarayne Thelas). It works either way and you get paid the same (100 gold).
Drarayne Thelas' house is right at the end of the center bridge. She has an infestation of Cave Rats (bigger, tougher versions of the common Rat). There is one rat in the bedroom of the house and two more in the upstairs storage area. You must kill all three. Drarayne will give you the key to the storage area, but you can pick the lock as long as there are no guards nearby (costs a few golds if you're caught).
The rats have 23 HP and do 1-10 damage with 50% combat skills (common rat has 23 HP and does 1-2 damage with 30% combat skills).
After the rats are dead, collect your fee from Drarayne and report back to Eydis to finish up.
There are two poachers in the Shulk Egg Mine, just down the river from Balmora. Your job is to get rid of them. There are a couple of miners just outside the mine entrance and a few more in the Miner's Camp section of the mine.
You'll encounter standard Leveled Egg Mine creatures on your way to the poachers in the Queen's Lair. If you talk to the two miners outside, they will tell you to feel free about killing any Foragers or Warriors that you encounter, but they ask you to leave the Workers alone (Workers will generally leave you alone as long as you leave them alone).
Report back to Eydis for your payment (100 gold and 4 Quality Restore Fatigue).
There are four Telvanni hirelings in a cave just north of the Caldera Ebony Mine complex. They are all Thieves Guild members (which may have an effect on your membership in that faction) working for House Telvanni. Eydis assures you that the agents are not sanctioned by the Thieves Guild. So head up the road to Caldera, go through town and take the first left to get to the mine, which sits in a valley. If you want to take the shortcut, simply levitate over the mountains west-southwest of Caldera. It's in the next valley. There are two parts to this mission: take care of the guard outside and take care of the three inside.
It's not too difficult to simply sneak up behind Alveleg and take him out quickly. If you are Thieves Guild member or plan on joining the Thieves Guild, your job is a little tougher. You cannot simply attack faction members; you must let them attack you first. Rather than resorting to the tried-and-true Taunt (unless you're really set on working on your Speechcraft), just talk to him about "Telvanni Agents." He should immediately end the dialogue and attack you.
Inside is a different problem entirely. NPCs in cave complexes are usually strung out so that you can take them on one at a time. Not here. All three are together and you'll have to fight them together. On the bright side, they are all ready to attack you on sight, so you don't have to worry about them greeting you like a long-lost cousin or offering you the standard NPC dialogue options.
Elsewhere in the cave you will find a bunch of crates and barrels to search (nothing terribly exciting, but it can generate quite a few golds), plus a trapped chest at the bottom of the very deep pool in the next chamber. You'll have to fight a couple of Slaughterfish to get to it, but it has a few golds, some expensive clothing and an amulet.
Report back to Eydis for your reward (400 gold).
Another quest that can put you on the bad side of the Thieves Guild. Sottilde over at the South Wall has a code book that the client wants retrieved. There are two conditions under which you can do the quest:
Sottilde will tell you that the book is evidence on the Camonna Tong's skooma smuggling activities and the Guild is trying to decode it so that it can be turned over to the authorities and the Thieves Guild can take over the business. She will also tell you that the Fighters Guild is working for the Camonna Tong since Hard-Heart took over.
You don't get any of that information.
There are three ways to get the book:
If you go talk to Percius Mercius at the Ald'ruhn Fighters Guild, he will tell you that it shouldn't be too difficult to just talk her out of it. My experience has been that Sottilde generally has a pretty decent disposition toward the PC, so it's only a matter of a couple of small bribes to get her disposition up to 70 so that she will just give it to you.
There only immediate repercussions for turning the book over to the Fighters Guild is that your rep in the Thieves Guild will drop by 10 points. The reward is only 50 septims (don't try to hold out for more or her disposition drops 5 points for each refusal), which ought to cover the bribe it took to get Sottilde to give the book to you.
If this one doesn't raise a few questions, you're just not into the spirit of things. Your mission is to collect 200 gold From Helviane Desele at Desele's House of Eartly Delights in Suran. If you go talk to her about it, she'll tell you that it's for "protection" from the Camonna Tong and will refuse to pay you unless she really likes you. Talking to Mercius about it will get you essentially the same information. He suggests that you just pay it yourself since it does not sound like a legitimate contract.
If you can get Helviane's disposition up to 75 or more, she'll pay the money and there does not appear to be any immediate repercussions for collecting the money from her. Your fee for being the heavy is a whopping 100 gold. Try not to spend it all in one place.
The outlaw Dura gra-Bol is living in Balmora and has a 250 gold bounty. The dialogue says "he", but it should be "she". You'll find her in the house at the foot of the southern bridge, on the 3rd floor. She has no dialogue options that would drop her into fight mode, but there aren't any witnesses, so you should be able to just take her out.
There are a few goodies laying around her house and you can pretty much help yourself after you've disposed of her. Report back to Eydis for your 250 gold.
This quest is not available until you are at least a Protector in the guild. Asking about "orders" before then will get you a polite referral to the other guild stewards.
A contract direct from Duke Dren himself. Orc outlaws have been spotted near the farm of Alof the Easterner. Go talk to Alof and take care of it.
Alof's farm is north of the Arvel Plantation and pretty much northeast of Pelagiad. Alof tells you that the Orcs are coming from the Daedric ruin called "Ashunartes". Alof's directions tell you that the ruins are northeast of his farm and west of an old Dunmer stronghold (Marandus, but he doesn't tell you that). He suggests that taking out the leader (Burub gra-Bamog) should be enough to make the others go away.
Alof's directions are pretty accurate, but he neglects to mention how to get over/around/through the mountains immediately northeast of his farm. You're the adventurer. Deal with it. There are two entrances to the shrine. One in the Daedric ruins proper and a second one just west of Marandus. One takes you into the second floor of the shrine (which can be really good if you have ranged spells or weapons and it's also where the leader is) and the other takes you to the main floor of the Shrine (good if you need to melee or want to take out the other three Orcs).
The ruins are practically swarming with critters:
Unlike most other Daedric ruins, though, there are no NPCs wandering on the outside, so if it moves, you can kill it.
Inside the Shrine are four Orcs. Although you only need to take out the leader, you'll most probably want to take out all four.
There are some decent goodies in the shrine proper, but the pearl in front of the statue is cursed (summons a Dremora Lord, who always appears behind you).
Report back to Eydis for your reward (500 gold and a 10-point boost to your Fighters Guild rep).
Similar to the previous quest, but without the official blessings of the Duke, you are to find and take out Dovres Verethi, a smuggler. Please note that the quest is to take out this one person. What Eydis doesn't tell you is that you're going to have to wade through six other NPCs to get there. On the bright side, if you are careful, you can take them out one at a time.
The smugglers are holed up in a cave called Mannammu, located southeast of Pelagiad. Don't take that too seriously. It's directly south of the Fort and facing north on the shoreline. If you're an intrepid adventurer, you may have already cleaned out this place while you were wandering in the area. Although it twists and turns quite a bit, the cave is linear and it's very difficult to get lost inside.
There is an underwater portion of the cave in which you'll find a couple of minor water creatres and a Dreugh, but there is no reason why you'd need to go there unless you're just being thorough.
Notice that there is really nothing to get excited about on the NPCs. Get excited anyway. There are 28 barrels, crates, and chests demanding your attention, plus a couple of pieces of steel armor, a couple of potions, a scroll, a book that increases you Mysticism skill and a book that increases your Security skill. Leave lots of room in your inventory before coming here, or plan on making several trips. Maybe both.
Report back to Eydis to collect your 1000 gold.
A Hunger is loose in the Sarano Ancestral Tomb and has stolen a helmet. But this is not just any Hunger. This is truly the wimp of all Hungers. It has no spells and only has 50 HP, though it still has its immunity to normal weapons. Hopefully you've picked up something decent by this time.
The directions aren't very specific, but the tomb is right next to the mountains north-northwest of Alof's Farmhouse.
The hard part is the six leveled tomb creatures that are also wandering around the place. If you're direct (take the first right), you'll only have to deal with two and the Hunger. The rest are farther into the tomb. On the bright side, the Hunger has an Ebony Helmet (you don't have to return it) and there is a Dreugh Shield behind a rock in the same room with the Hunger, so the trip won't be a complete waste.
If you're adventurous (and you must be or you wouldn't be here), you'll find four chests, a bunch of ingredients and an iron war axe elsewhere in the tomb.
Your reward is 1000 gold and this is Eydis' last guild quest. You will probably run into her again as part of another Fighters Guild quest and as part of a Thieves Guild quest.