Most spiders do not have common English names. I find it easier to remember the latin species name if I know what it means.
Bathy - "deep"
Bathyphantes - Deep Weaver
Bathyphantes approximatus - "close to" - Similar Deep Weaver
Bathyphantes gracilis - Slender Deep Weaver
Bathyphantes parvulus - Small Deep Weaver
Bathyphantes setiger - Setiger's Deep Weaver
Leptos - "thin"
Lepthyphantes - Thin Weaver
Lepthyphantes leprosus - Skinny Thin Weaver
Lepthyphantes major - Greater Thin Weaver
Megalepthyphantes - Giant Thin Weaver
Megalepthyphantes collinus - Collin's Giant Thin Weaver
Megalepthyphantes nebulosus - Misty Giant Thin Weaver
Phantes - Greek, "weaver"
Segestria florentina - blanket or counterpane, a reference to the shape of its web outside the burrow, from Florence - Florence Blanket Spider.
Tenuis - "thin/slender"
Tenuiphantes - Slender Weaver
Tenuiphantes alacris - Brisk/Quick Slender Weaver
Tenuiphantes cristatus - Created Slender Weaver
Tenuiphantes flavipes - Yellow-footed Slender Weaver
Tenuiphantes nigrinus - Black Slender Weaver
Tenuiphantes zimmermanni - Zimmermann's Slender Weaver, after Walter Max Zimmermann (1892–1980)