I understand that the regular clones speak in third person from the answer here. But Last Order actually has emotions, so there is no reason to speak in third person or repeating her name over twice. What is the true reasoning behind Misaka wa Misaka wa ...?
Is there a canonical answer for this?
Last Order isn't just saying her name twice. She narrates herself, like all the SISTERS, but she is doubling her dialogue tags (quoting herself twice) in her narration. This isn't just simple yakuwarigo but rather due to in-universe lore reasons.
The normal SISTERS narrate everything they say because they are always logging everything they do to the MISAKA Network and the narration is there to provide context so the others can understand their situation better. The MISAKA Network is also why they speak and think in third-person: because 10000 different streams of data that are like "I did this thing." floating around in the Network at the same time is confusing.
As a side note, the SISTERS are fully capable of speaking normally. In fact, they do not narrate themselves if they are doing something which they don't want the rest of the Network to know about. This is why Last Order sometimes drops her speech pattern during intimate moments with Accelerator, she wants to keep those events to herself.
Anyway, back on topic, Last Order is unique in that she is also the system administrator. As such, anything she broadcasts to the Network could potentially be interpreted as a command to the rest of the Network. In order to differentiate the messages she logs to the network where she is simply transcribing her own activities as an individual from messages where she is communicating directions to the rest of the Network, she puts the former type of message inside a quotation, with the implication being something like "this is Last Order uploading as an individual node and not as the Admin of the Network". Thus when she is logging her own speech, she ends up having double quotations.
Basically, "MISAKA wa MISAKA wa" + "mitari" appears to function as what in programming terms is called an escape sequence.
Now let's demonstrate this with an example. https://youtu.be/401oSrxp33c?t=82
Last Order's dialogue is as follows: Anata no heya tte doko tte MISAKA wa MISAKA wa shitsumon shite mitari
There are three components to this line of dialogue:
The escape sequenceTranslated in such a way as to preserve the distinctness of all three components without bothering with making it flow well, it would be something like: MISAKA appears to express Where is your room questions MISAKA
The dialogue of normal SISTERS lacks the third component, the escape sequence. If a normal SISTER were to ask this question, it would be something like: Anata no heya wa doko to MISAKA wa shitsumon suru
Translation: Where is your room questions MISAKA
Addendum: There is a third speech pattern among the MISAKA SISTERS, which is that of MISAKA Worst.
MISAKA Worst does not narrate her dialogue but she still speaks and thinks in third-person. This is because she is only partially connected the MISAKA Network. Worst is something like an Observer, she receives information from the Network but she cannot upload anything (therefore she does not need to narrate herself) nor is she subject to commands from the Admin.
If Worst were to ask the above question her dialogue would be no different from normal speech: Where is your room?
To understand why she speaks like this, we need to understand the grammatical rule for "wa" in Japanese. I'll admit that I do not understand Japanese fluently and I am a beginner, if anything, but let's take a look at what it can mean. For reference:
http://japanese.about.com/library/weekly/aa051301a.htm
The first one implies that the "wa" is a subject marker and she using herself as the subject. She has emotions unlike her sister clones and her having emotions mean that she can feel a variety of human characteristics. What I'm saying is that it is possible she repeats her name in this manner to draw attention to herself, to show that childish selfish characteristic. And we can see that she sometimes has no consideration about what other people feel, like how she (seemingly) unintentionally refers to Yomikawa as an old woman in one the bathroom/shower scene. Although this can be explained as her being playful, but I believe this still reflects her childish selfish nature, not that she's actually incapable of feeling empathy.
"Wa" can apparently be used as emphasis for the subject at hand, so this gives more credence to my theory. From my reference:
"Beside being a topic marker, "wa" is used to show contrast or to emphasize the subject."
Although I can be wrong on this one.
"Wa" as a contrast does not work well as it does not explain much.
So it may be that she repeats herself in this manner to grab and draw attention to herself much like a child would. Aside from her being just a child, perhaps she does this to distinguish herself from the other clones.
I would say it is definitely a character trait or characteristic rather than a language rule or regional dialect.
@Frosteeze is on the right track with Last Order. I think it is a reflection of either her eccentricity or her childish behavior. There is a long history in anime/manga for eccentric and mischievous 2nd line characters to have some unique and often cute verbal tic.
In Shakugan no Shana, Wilhelmina ends nearly every sentence with "de arimasu".
In Saki, you have Yuuki's habit of ending her sentences with "d'jey" and sometimes slipping into a 'hick' accent (i.e. Hokkaido or Okinawa). In fact, half the cast seems to have some sort of distinctive catchphrase.
You get the idea. 'Tis is a common trope.
