EDITED IN:
The ones with [COMPLETE] are the ones I'll not change.
Eevee > Vaporeon > Oceaneon [COMPLETE]
Eevee > Jolteon > Ampeon [COMPLETE]
Eevee > Flareon > Exploseon [COMPLETE]
Eevee > Espeon > Helion (named for the Greek word for "Sun") [COMPLETE]
Eevee > Umbreon > Seleneon (named for the Greek moon goddess) [COMPLETE]
Eevee > Leafeon > ??? (OK, this line DON'T have Cholorphyll.... name ideas?)
Eevee > Glaceon > Greleon (If Andcetera is correct, "Grele" is French for hail...) [COMPLETE]
Eevee > Kungeon > Masteon (fighting-type line)
Eevee > Toxiceon > Venoeon (poison-type line, 2nd one is based a well-known spider-foe, cause he's pure awesomeness!)
Eevee > Siroseon > Terreon (ground-type line) [COMPLETE]
Eevee > Cloudeon > Cumuleon (named after the cumulonimbus clouds, A.K.A. thunderheads. this line is the only pure flying type one.) [COMPLETE]
Eevee > Insecteon > Arachneon (bug-type line, any better idea for names?)
Eevee > Stoneon > Diameon (rock-type line)
Eevee > Poeon > Phanteon (ghost-type line, based on Legend of Zelda foes) [COMPLETE]
Eevee > Drakeon > Zilleon (dragon-type line, & Drakeon is one of Jezza's pokemons, & it's evo is named after Godzilla!)
Eevee > Oreon > Alleon (steel-type line)
(Should I add a normal-type line?)

Kungeon And Masteon?
What do those mean?
Leafeon could be...... Floreon (though the flower idea seems overdone with)

A new type has been announced for Generation 6.It is the fairy type eeveelution named as sylveon,the lightest eeveelution.

Slyveon sounds weird to meh.

Just for into:Fairy type is strong against dragon type and weak against fire and steel type.

I know dat.

I want a rock type eevee family member.
Name would be Bouldeon

And Dark type?
Dazeon (Night Daze)

Dragon Type
Draceon

Ghost Type:
Haunteon XD

Poison type
Sludgeon

I don't know more.

Then I know
Dugneon.
Ground type.

Eartheon would suit.

Its upon you what do you name.I got Dugneon and you eartheon.
Next would be ghost type.
No idea of its name.