I believe battojutsu was the predecessor of Iai. I don't know about the term but I believe the first Iai Ryu was Tamiya in the early 1600s. Tamiya Shigemasa served Tokugawa Yorinobu and was a disciple of Hayashizaki Shigenobu who I understand to have started battojutsu. However, like many ancestral records from this time......best to have some scepticism.

just to add, the term is from the Warring States period (戦国時代) - from my understanding it didn't exist before this time.

I'm pretty sure that I've come across the term Iai (居合) prior to Tamiya-ryu's founding, but it had nothing to do with batto, or rather sword drawing methods; we can still see the term iai compounded outside of the martial arts, though I can't think of any examples at the moment...

And of course the sword drawing methods, regardless of it's name, existed for quite a bit longer, but I also suspect that it becoming a discipline per say, began with Hayashizaki Shigemasa.

That's interesting - let me know what context you find the term outside of martial arts. Is it a spiritual or religious context? I'd be fascinated to know.

居合わした is a term that may have some relevance.

I haven't come across this myself but it appears in Weblio.

I'm currently working on a course through Stanford where I'm directing my research on this very subject, so we'll see what turns up.

Is it cheating to use wikipedia? The information checks out with the few books I have on iaido and aikido, for what that's worth.

http://en.wikipedia.org/wiki/Iai

The term 'iaido' appear in 1932 and consists of the kanji characters 居合道.[9][10] The origin of the first two characters, iai (居合?), is believed to come from saying Tsune ni ite, kyu ni awasu (常に居て、急に合わす?), that can be roughly translated as “being constantly (prepared), match/meet (the opposition) immediately”.[11] Thus the primary emphasis in 'iai' is on the psychological state of being present (居). The secondary emphasis is on drawing the sword and responding to the sudden attack as quickly as possible (合).

Last character, 道 is generally translated into English as the way. The term 'iaido' approximately translates into English as "the way of mental presence and immediate reaction",[9][12] and was popularized by Nakayama Hakudo.[9]

The term emerged from the general trend to replace the suffix -jutsu (術?) with -dō (道?) in Japanese martial arts in order to emphasize a philosophical or spiritual aspects of practice.[9][13]

A BRIEF HISTORY OF THE ART OF BATTŌ-JUTSU:

The term 'battōjutsu' (抜刀術), or 'nukutō no jutsu', literally translated as 'the art to extract the blade', indicates the practice which consists of drawing the sword and to cross, avoid, or strike, all within in a single movement, without the enemy being able to see or feel one's initial intention. Relations between the various traditions of battōjutsu and the first three sword traditions of Japan (Nen-ryū, Kage-ryū, & Kashima-ryū), postulate that this art existed already within these three founding sword schools. Many chronicles describe such sword luminaries as Bizen No kami, Bokuden, Hidetsuna, & Muneyoshi, as well as their disciples, as excelling in the art of iaijutsu. However, the techniques were a collection of various and often vague principles for using the body that led to a freedom of interpretation.

The first tradition which did specialize and codify this art into a precise methodology was founded by Hayashizaki Jinsuke (1542 -?) at the beginning of the Edō period. Hayashizaki transmitted his method to only three disciples. One would go on to become his successor and the other two would eventually found their own traditions. During second half of the Edō period, the schools resulting from Hayashizaki-ryū used the term of 'iai' (居合), rather than the term of 'battōjutsu'.

There exists a score of terms which all are read as ‘Iai’, but are translated broadly as the action of 'engaging and fighting an approaching enemy'. ‘Iai’ means 'to link' and 'to be', which can therefore be translated as 'linking the intention and the movement in a moment when the technique must be carried out'. The large majority of the techniques of iaijutsu, found in traditions born during the second half of the Edō period, were practiced starting from sitting positions, where the movements were extremely restricted.

- DR. KACEM ZOUGHARI (Was a Facebook post which I cannot find now).

"Iai (居合) means to match something, someone, an action, according to the position that you sit. From where you stand, sit, walk, that position, that body structure, with the weapon you have, you have to match the attack of someone. It doesn’t mean cutting, most of the people who translate Iai interpret it as “the art of the fast draw,” or fast cut-draw. It doesn’t mean this. First Iai means to match the situation, according to a certain situation, from the position where you stand. That’s for the modern word, Iai.

The old word for this was Battō (sword draw; 抜刀), and here it is already more logical. Nuku (draw; 抜く), Tō (sword; 刀). This is also connected to the technique in Kotō-ryū that we practiced last night, Batsugi (skill of drawing out; 抜技). There is also sayahiki (pulling the scabbard; 鞘引き), hanare (separate; 離れ), Sanbyōshi (three cadences; 三拍子), which is for example for Kotō-ryū. So each school has its own word and terminology. Sometimes it is in order to express the difference, sometimes just because this is the way.

Iai is also in the nine schools, in ninjutsu, but also in the classical schools to. Iai is also considered like a Kyūgoku Ōgi (級極奥技), in Japanese, Ōgi (奥技), gokui (極意), is like the highest level movement, the highest level technique, the highest level everything. Each school has that. And kyūgoku (級極) means extreme. So it is the highest level to the extreme."

Source: Zoughari, Kacem. margate Shinden Fudo-ryu Shoden and Gyokko-ryu Iaijutsu (2011). 

http://onmitsukage.com/programs/margate-2011-shinden-fudo-ryu-and-gyokko-ryu-iaijutsu

Conference Paper Margate 2011 Shinden Fudo-ryu Jutaijutsu and Gyokko-ryu Iaij...

The source which gives the first attestation of Japanese words is the dictionary Nihon Kokugo Daijiten (just as for English it is the Oxford English Dictionary).  According to the NKD, iai was developed around 1570-1592, but the word, with this meaning, first appears in 1646.  The same word, with a completely different meaning is attested from 1195. 

Thank you Mr. Lawrence, I have not yet had access to the Nihon Kokugo Daijiten, I would be very curious as to that 1195 definition.

the meaning associated with the 1195 word (same pronunciation and Chinese characters) is something like "accepting a quantity that is in a written record without actually going out into the field and measuring it".  The  i of iai originally meant 'sit', and the -ai has the meaning of the word means 'together'. so it meant 'bring together while seated'.

Thank you again Mr Lawrence, that is again very interesting!

Much food for thought, especially considering what modern Iaido looks like (both in reference to the tendency to operate from seated positions, as well as the "accepting a quantity that is in a written record without actually going out into the field and measuring it").

Considering the (what I perceive) relevance of this older definition of Iai and it's use in the Edo-jidai, it almost seems like Batto (抜刀) and Iai (居合) grew on two different timelines and met later (if that makes any sense). Like the idea behind both terms (and thus how they influence the practice) suggest a different focus in it's application. Though this could be a little ridiculous to connect appropriately, yet here we have the two terms currently meaning pretty much the same thing.