A variation on Trexor above that sounds and feels a little more upbeat:
Trexa (Trade / Exchange / Autonomy)
my proposal: "xbits"
x for exchange
bits for bytes.
Just saw “bit” was frowned upon.
An alternative…
“Cyberia”
Thank you all for the numerous proposals so far. Since the launch of this campaign mere five days ago we’ve collected a whooping 1630 entries (and counting)! Well done everyone!
As promised, throughout this campaign we will share the names in our top list among those we’ve brainstormed internally with the team. Your proposals already coincide with 4 of our top names, so these will be released first. And here they are:
Bit² - only a cosmetic change to Bitsquare, keeping the original sound. Unfortunately this cannot be registered, as it still contains the (sound of the) term “square”.
Bit³ (bitcube) - an “upgrade” to Bitsquare, getting rid of the problematic term. However, it is used a lot (55k results) and as such it is not unique. We might just be replacing one problem with another.
Bitflow - Conveys well the flow of (digital) value - a core principle in Bitsquare. However, it is also used too much (771k results) and public records show that a TM registration with that name has failed in 2015.
Bitswap - Another take on the exchange of value. This time it is being swapped between users. The name is used here and there, most notably for an open-source protocol. In that case we’d rather not interfere.
As you might have noticed, all four of the above names start with the term “bit”! Although we have nothing in particular against this term (!!!), it appears to be overused in the crypto space. Also 50% of all proposals so far contain the word “bit”.
Therefore, we would like to ask the community to please refrain from using “bit” in new proposals - we simply have enough of those. If we do end up adding “bit” to a name that one of you has proposed, that person will still be considered the winner. Note that all existing “bit” proposals will still be considered and are eligible for the bounty. We would now like to promote diversity. 
For further inspiration, let me end with the core values that Manfred summarized earlier:
Openness, protection of privacy, permission-less, alternative structures for organizing and funding OS projects (DAO), synergy, resilience, value transfer, new more flexible and open forms of money (value representation),…
A name representing some of the above would be a great candidate.
Lastly, please do discuss current proposals - we’d like to hear what the community thinks! But remember, although constructive criticism is encouraged, criticize ideas, not people. You can praise ideas too! 
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Widebit(s)
(Bit)broad
Broad(bit)
(Bit)draft
(Bit)ace(s)
Btrigger
Trigger(bit(s))
Te(t)rabyte(s)
Terra(bit(s))
(Bit)axis
(D)Exponent
B(i)tom(s) (… after Atom…)
(Bit)gene(s)
Genebit(s)
Genobit(s)
Genepool (collective human DNA)
Cryptogene(s)
Cryptonome(s)
Cryptonoma
Genoma(s)
Genome(s)
Cryptodira (looks safe and solid to me they managed to evolve since Jurassic period afterall though I grant you the one on the picture seems to be sleeping or dead…)
(World)WideWorthy(-ies)
Compeer(s)
Peerite or
I do like the “fool’s gold” notion much.
or
Pyrit
ClearCut
Dexwap
Cachex
BandyBuffer
Tradebuffer
SwapSpace
(D)Execho(es)
(Bit)Even
Electronlibre
Agoraflowbia
Agoraflow(bits)
Pure brainstorming exercise around the words peer, exchange, open, global, market… Sorry for the long list, I hope you can find some inspiration.
Peertocoin (Peer2coin)
Peercoinplace
Peercoinplaza
Peertotrade (Peer2trade)
Peercoinswap
Smartpeer
Smarterpeer
Peersmart
Peerstorm
Peerpurse
Globalpurse
Globalpulse
Pulsereactor
Peerpulse
Peereactor
Peerflux
Peerflex
Purepeer
Tradepeers
Tradingpeer
Tradetopeer (Trade2peer)
Peertrade
Peertradex
Peerkeep
Metapeer
Multipeer
Multitrade
Tracelesstrade
Tracelesstradex
Betterex
Easypeer
Easycoinpeer
Trulyflexypeer
Truflex
Peerflex
Flextocoin (Flex2coin)
Flexchange
Openex
Opentopeer (Open2peer)
Opencoin
CoinOp
PeerCoinMarket
Peerplace
Peerplaza
Peerswap
Freepeer
Hidex
Trueque (Spansh word for barter)
What is Bitsquare for? We are just exchanging digital tokens as people exchanged eggs for meat in the past.
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As my username suggests, I’d like to propose the new name, and subsequent URL for BitSquare to be https://btc.me/ , aka BTC.ME . I think it’s simple enough for someone to pitch it in a hurry, and memorable. At the end of the day WE make up the Bitcoin community, and the more we cultivate the space, on an individual level, but also within our communities, the greater the adoption will happen. Explaining Bitcoin to someone new, can be long-winded sometimes, as there are many critical points to get across. With hope, this emerging platform can simplify the process of attaining Bitcoin and other crypto-currencies, and get people on the path to blockchain-righteousness. I haven’t registered the URL, or anything like that, so if you REALLY want it, it’ll be easy to do so. Aloha!
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MOSAIK :
like a great mosaic made of various pieces (bitsquare’s core values).
AREQUBITS :
anagram of the word bitsquare.
WYCAL :
it means “only What You Can Afford to Lose”
Don’t forget to register “liberab.it” as well…
Decorum
A decentralized exchange is the appropriate exchange style to the subject of decentralized money.
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I agree about the word “bit” being overused and sounding fairly generic. Now, HyperCube on the other hand sounds pretty cool eh?
Anybody love it or hate it? I’ll browse and see if any suggestions jump out at me.
Interesting! Thanks for the suggestion!
AutomEx
A decentralized exchange is just an exchange automated enough to get rid of third parties. There is a car dealer called Automex, but I guess there is no conflict given the different markets.
It should flow in a euphonic manner as much as in meaning.
Puquio PU-key-oh
The Puquios are an old system of subterranean aqueducts near the city of Nazca, Peru. They were relied upon to bring fresh water into the arid desert. The Puquios have never been fully mapped, nor have any been excavated.
Estuary
Estuaries form a transition zone between river environments and maritime environments. They are subject both to marine influences—such as tides, waves, and the influx of saline water—and to riverine influences—such as flows of fresh water and sediment. The inflows of both sea water and fresh water provide high levels of nutrients both in the water column and in sediment, making estuaries among the most productive natural habitats in the world.
Decoin
Get rid of one kind of decentralized & dematerialized coin for another, in a decentralized & dematerialized way.
CoinWizard
BitWizard (Wiz)
BlockWizard
CoinMagic
Magex
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Evenflow
Flowrider(s)
Flowrunner(s)
SpeeDeX (or SpeedX, or SpeeDX)
Fast decentralized exchange. There is a company called Speedex, but unrelated to currency exchange (it is rather a Fedex-like company).
TrustlEx
Trustless Exchange.