I'd be particularly interested in who first used the name in a European language and whether it was used in a non-European language such as Arabic, Persian, or an Indian language before that.
[Edit 2010/01/22: Thanks to everyone who responded. It took me a few days to check Jonas Meyer's references. (The discussion of the CRT is on pp 175-176 of Part III of Wylie's book.) As JM said, they seem to narrow the appearance of the name in a European language to 1853-1929, which is hundreds of years later than I expected, and it now wouldn't be so surprising if it first appeared in English, maybe even in Dickson's book. So,
Question: Are there any European languages in which the CRT has a name that is not a direct translation of "Chinese remainder theorem"?
One more point: Wylie says,
'In examining the productions of the Chinese one finds considerable difficulty in assigning the precise date for the origin of any mathematical process; for on almost every point, where we consult a native author, we find references to some still earlier work on the subject. The high veneration with which is has been customary for them to look upon the labours of the ancients, has made them more desirous of elucidating the works of their predecessors than of seeking fame in an untrodden path; so that some of their most important formulae have reached the state in which we now find them by an almost innumerable series of increments. One of the most remarkable of these is the Ta-yen, "Great Extension," a rule for the resolution of indeterminate problems. This rule is met with in embryo in Sun Tsze's Arithmetical Classic under the name of Wuh-puh-chi-soo, "Unknown Numerical Quantities," where after a general statement in four lines of rhyme the following question is proposed: ...
In tracing the course of this process we find it gradually becoming clearer till towards the end of the Sung dynasty, when the writings of Tsin Keu-chaou put us in full possession of the principle, and enable us to unravel the meaning of the above mysterious assemblage of numerals....'
The Song dynasty apparently ended in 1279, which gives an interval of several hundred years. So, it seems that the name Chinese Remainder Theorem is not completely unreasonable, since according to Wylie, it's not clear when the general form was discovered, or at least might not have been at the time the theorem got its name. ]