Mumuth :D
Thought it flows well and it's a play off of "mammoth". :)
(I haven't read all the posts, so sorry if anyone else has said this one already which is always possible when lots of smart and creative people are tossing out ideas. lol)
I dont know why but I came up with: Firemite
[IMG]http://i195.photobucket.com/albums/z35/MonicaRafael34/96095.jpg[/IMG]
Ki-Rin or Qi-Rin, or some variant.
Reason:
https://en.wikipedia.org/wiki/Ki-rin_%28Dungeons_%26_Dragons%29
Love is all you need.
Just based on there not being any pet species with this starting letters:
Umanth (From Mammuthus) Uskon (Play on "tusk" and "lion")
Alparslan - literally means heroic lion in turkish. name has some cool history Nalvay Kadivai Airavata - mythical white Hindu elephant that carries the god Indra
BAKUDON!
I would name it Bakudon! (Like mastodon mixed with baku!)
Reason: Baku = furry trunk nosed dream eating spirit Mastodon = awesome extinct elephant relative
[size="8"] [tot=Hazard] Looking to adopt four letter english noun Legacy Names like Glue, Disc, Lash, Crop, Gown, Teas: to make characters.[/size]
Airavata (In Indian mythology it is the creator of clouds...since shengui goh has peaks in the clouds it makes sense and sounds very airy like the color palette is.)
Usuku. yew-shuu-kew. Combination of the Japanese word Usucha, meaning "thin tea" and Baku a Japanese spirit that ate/drank dreams and nightmares!
Bakura was the first thing to pop into my head, a lot of people are using Baku the dream eating elephant god I noticed doing a quick skim through a couple pages XD Which was the same thoughts I had.
A few ideas, most of which have already been said I'm sure:

Yera
Yoloti
Uli
Urbain
Quennell
Quinzell
[img align=right]https://i.imgur.com/bvrm5Sg.jpg[/img][/align]
Activities
Qiphant
RIP Mom, I love & miss you more than you know. Tell Dede hi.
Ocean Conservation Namibia on YouTube... they do good work! https://www.ocnamibia.org
Xiang-Shen Xiang-hun?
They're both related to spirits, but translate to different types.
xiang is the word for elephant in chinese Shen is deity Hun- soul,
Liphan- just cause I like the sound of it.
Qiang- Qi being an alternative spelling of Chi (spirit energy), and Xiang being Chinese for elephant.
This might be too on the nose and someone may have already suggested it, but what about
Phantasy - combining elephant and fantasy
Oh man, it's an elephant isn't it. I kind of wish it was like a saber tooth tiger or something, but that's still okay.
Naming suggestions...
Sabal and Amita, come to mind because of Far Cry 4, which took place in a fictional part of India, and riding elephants was a nice gameplay thing you could do there, as well as the two of them being referred to as "the two headed elephant" of their rebel team.
But maybe the name Bhadra would be better, the girl who was supposedly the human embodiment of the land itself. A much kinder and wiser character than the two mentioned, though ultimately one who does not end up with a kind fate no matter the choices you make.
Or maybe the name could be some sort of mixture of all three.
Samita Bhradal Asadhral
Though in a more Chinese line of thinking, which would be pretty cool because there's a lot of Asian themed things but I'm always carving more representation for my Chinese heritage, a name that could reflect that would be really cool.
Mèng, part of Mèngxi?ng, meaning dream, and xiàng meaning elephant.
Since the second half of dream sounds similar to the entire wording of elephant, I thought it might be neat to use Mèngxiàng as the possible name. Dream elephant, Mèngxiàng. Pronounced like mung-shao.
Taroak - "T-air-oak" - Base concept was tarot cards Manari - "Man-ar-ee" - Base concept was mana Krystalia - "Kris-tawl-ya" - Base concept was crystal Zeerek - "Z-ear-ek" - Base concept was seer
I tried to describe the pronunciations the best I could.