Where does Grendel's mother live?
Grendel's mother lives in a cave beneath what is called a mere. A mere is a 12th Century term for a standing body of water. The Old English term can mean a lake, but given the amount of time it takes Beowulf to swim to the cave at the bottom, the usage meaning sea-like lake might be closer to the original intent. This means that the lake may have been an arm of the sea coming inland. The fact that the lake was within a "fen" would also mean that the mere would have to be an arm of the sea. A fen is an Old English term for a marshy, frequently flooded area of land. Unlike a swamp, a fen does not have woody trees. It is full of grasses and reeds that are fairly low to the ground. Fens tend to be foggy and dark, matching the description of it as being forbidding.
The mere was quite a distance from Heorot. It says that they must cross a moor, over steep stony slopes, narrow ways, choked paths, gullies, cliff ledges, and haunted lakes to get there.
The area surrounding the mere is referred to as a mountain grove. Above the mere is gray stone with a hanging wood. The water of the mere is dark and dour - meaning rocky and infertile. The water seethes with blood. The use of the word seethe is interesting because the word requires movement such as a boiling action or an agitation. The water is also referred to as blood red, as the soldiers are waiting for Beowulf's return.
The surroundings add to the gloom. They find Aeschere's head next to the cliff. And, as they watch the water, they see serpents, sea-dragons, water daemons, worms and slithering wild things going in and out of the lake.
Once Beowulf swims to the bottom (remember it took him almost a full day to get to the bottom) he is attacked and dragged into Grendel's Mother's cave. It is full of horrible tusked creatures that attack him. The cave has an area that keeps the water out and it is lit by firelight. Beowulf likens the space to a hall, so it is quite large. There is pile of weapons forged by giants at the edge of the cave and Grendel's corpse near the wall. (It is somewhat hidden because Beowulf has to search for it.)
Grendel's mother lives in a lair at the bottom of the swamp. In order to get to it, Beowulf must swim underwater for what seems like hours. Interestingly, this underwater lair is reminiscent of hell - with demons, creatures, and evil lurking all around.
When Beowulf reaches this underwater lair, it is a lot like a battle hall, but somewhat empty. The creatures that attacked Beowulf were mostly in the water leading up to this lair. The only important items that are present happen to be Grendel's mother and the sword forged by giants.
Grendel's mother lived in a cave. To get to it, Beowulf had to swim down into a lake. The cave is the inverted version of the hall where the men slept. It too is a battlehall, and full of meaningful trophies, but rather than being full of friendly people, it is full of strange, dark, and magical creatures, all of which assault Beowulf.