Ha good one!
Debate: What to call "Online Chess"...

Non rated chess, because some people will spend seconds on each move whilst others will spend hours.

"It's chess, Jim, ... but not as we know it."
"What is it, Bones?"
"It's the forum, Jim ........... It's --- Dead!"
Yes, all chess regardless of time controls can be played online. So "ONLINE CHESS" is ambiguous.
"LIVE CHESS" gives me the impression that both players need to be there to play the game. This is suitable for chess games with non-day time controls. In other words, those time controls used in OTB play.
"TURN-BASED CHESS" is silly; chess is not an action game! Chess is always turn-based.
Another related term is "REAL TIME CHESS". This conveys a similar meaning to "LIVE CHESS", but can be confused as chess played MMO style. Avoid this term.
"CORRESPONDENCE CHESS", traditionally refers to chess that was played through sending mails to each other. By itself, useable but probably not the best. People can't know without joining, whether you are an intermediary finding opponents for them and then they have to send mails to each opponent, or whether you offer them the option of playing games with day time-controls.
So, in my opinion, two terms should be used:
1. ONLINE LIVE CHESS, for non-day time-controls
2. ONLINE CORRESPONDENCE CHESS, for day time-controls (hyphenated to prevent ambiguity!)
Of course, in the end those terms will still need some explaining for some people. Maybe you should just create a page to explain the difference. The page where you promote your site is the best place.
EDIT: Better yet, juxtapose them. Put them side by side for the potential signups to compare.

Ah, good idea. Can split test using a free adwords voucher and see which name of the game "pulls" more response.
What's in a name?
I've heard it's how some successful authors decide on the names of their books and even the version of book cover they choose to use.
(For some odd reason they deem directly measurable and accountable stats from real marketing tests to give them more value than say, crowdsourcing or polling prospects with no skin in the game.)
Of course, any copy needed for sales is split tested (and multivariate testing, taguchi method, etc.) and a name can be just as marketable a product as any benefit it provides.
Of course Coca Cola may not have won the blind taste test against Pepsi Cola as the best tasting cola, but even so has remained the best selling cola.
It uses alliteration with similar sounding words that make up its name exploiting the phonological loop phenomenon of the human brain.
Coca Cola
What happened to Palm Pilot after they dropped the "pilot" part and lost the benefit of alliteration with their marketing naming strategy?
Well they got replaced by a product that uses the alliteration...
Black Berry.
Conspiracy?
Maybe not.
Maybe we can hash it out over a cup of coke and game of
Challenge Chess.
cha-ching.
Erik, with the extra zeros I just added to the site's revenues there in a few seconds (on the left side of the decimal) with a forum post, just think how smart a business move it'll be to retain an hour's consulting time with me.
But then again, could always just crowdsource with a forum poll.

A Few Suggestions :
1) Your Move or Mine Chess
2) It's Your Move Chess
3) Play at your Leisure Chess/Leisure Chess/Online Leisure Chess
4) Preferred/Selective/Optional/Variable/Flexible Time-Control Chess
5) Your Choice of Time-Controls Chess
6) Oh No ,It's my Move Again Chess
7) Move Before u Lose Chess
8) Don't Forget to Move Chess
9) You are about to Loose on Time Chess
10) Analyse n' Move Chess/Think n' Move Chess
11) Bondage Chess ( juz kidding ,but it's kind of .. lol)
12) Burdensome Chess ( also juz kidding ,kind of too .. lol )

Can't believe I'm actually going for the Gold --- again ...!
Super-Cyber-Chess
Silly-Cyber-Chess
Psilocybin-Chess (oops, wrong website...)
Signals-Sent-Skyward/Settling-Sinister-Silent-Seconds-Later Chess
Check-In/Check-out/Check-again/Check-mate!/(Cheesuz!) Chess
(Yep, you're all correct: I shouldn't be posting these --- this vein may actually be played out .)
Worldwide Conflict Happily Excluding Severe Sequelae