This use of next Tuesday to mean Tuesday of next week is fairly old, according to the Oxford English Dictionary. They include examples under the definition:
Applied (without preceding the) to days of the week, with either the current day or (in later use; orig. Sc.) the current week as the implicit point of reference.
Thus (for example) next Friday may mean ‘the soonest Friday after today’ or ‘the Friday of the coming week’. The latter may be indicated contextually, e.g. by contrast with this, but it is not always clear which meaning is intended
a1592 R. Greene Frier Bacon (1594) sig. B, Thou knowst next friday is S. Iames, And then the country flockes to Harlston faire.
So the use of next --day came after 1592. However, there is another use of next with a weekday that predates this:
c1390 Chaucer Miller's Tale 3518 Now a Monday next, at quarter nyght, Shal falle a reyn.
There is also this:
c1230 (1200) Ancrene Riwle (Corpus Cambr.) (1962) 211 Ȝef‥ȝe ne beon nawt ihuslet i þeose isette tearmes, beoð hit þe neste [a1250 Nero nexte] sunne dei.
Having traced back the references in the OED, it is possible that the form next --day is a fairly original structure. Next has been used to mean "Designating the time, season, etc., following directly after one described, spoken of, etc." in both early and late Old English.
The OED does note that next with a weekday is:
Applied (without preceding the) to days of the week, with either the current day or (in later use; orig. Sc.) the current week as the implicit point of reference.
So next Tuesday originally meant the Tuesday after whatever day today is, but there is no clear notation of when the additional meaning was added. The use of Tuesday after this current week was noted in the OED as being used by 1711.
Definition of overnight is "for the duration of a night" from ThefreeDictionary
"That will happen overnight on Wednesday"
It means this will happen on night between Wednesday and Thursday.
Best Answer
To me, “next Tuesday” means the Tuesday that comes next week. For example, on Monday, October 11 and Wednesday, October 13, “next Tuesday” means October 19. Whereas on Monday, October 18, “next Tuesday” means October 26. “This Tuesday” refers to the Tuesday that comes this week, which on Wednesday would refer to yesterday, and on Monday refers to tomorrow. Similarly, “Last Tuesday” is the Tuesday that came last week. Without any descriptors, day names by themselves mean the next such day in the future unless used in the past tense. “He will do it on Tuesday” means the next time there is a Tuesday, which on Monday would mean “tomorrow” but on Wednesday would mean the following Tuesday. In the past tense, day names mean the last such day which occurred. “He did it on Tuesday” means the most recent day that was a Tuesday.
Not everyone may agree with this analysis, but in my experience this is what most people mean when they use “this”, “last”, and “next” with day names.