I looked at the FIFA website and there is a section called Milestones under the Laws of the Games section.
http://www.fifa.com/refs/laws/milestones/index_E.html
On the main page, it gives some very general milestones from 1580 through 1937-38. Then it also gives a list of amendments passed to the laws broken down by decade for the 1970's, 1980's and 1990's.
In the 1970's there were no amendments regarding substitutions.
In the 1980's there were are couple mentions of substitutes (in 1981 and in 1986) but nothing about a change in the number. 1981 mentions allowing a substitute for an injury. 1986 just clarifies at what point a substitute is considered a player.
Does this mean that prior to 1981 you couldn't substitute at all but beginning in 1981 you could substitute for an injured player?
Then in 1988 we have this:
Law III - Number of players:
Substitutes may be used under the rules of any official competition under the jurisdiction of FIFA, Confederations or National Associations, ...
...a team shall not be permitted to use more than two substitutes from more than five players...
Does anyone know what changed here? Was this the change that now allowed up to 2 substitutes for any reason. Also, these subs must come from 5 named players, so the roster size at that time must have been 16. Or had you been allowed to use 2 subs prior to 1988 and this was simply limiting the game day roster size?
Then in 1994, the law was modified for 2 + 1 -- A team may also use a 3rd substitute provided that substitute is for the goalkeeper.
Then in 1995 (as Alberto pointed out), the law was modified to allow up to 3 substitutes without restrictions to position.
There is no mention in these abbreviated amendments when the roster size was expanded from 16 to 18, but in 1996, there is a change that says that the rules of competition may state how many substitutions are allowed, from 3 up to 7 -- so there was apparently 18 players permitted at this point.
Anyone aware of any information prior to the 1970's or perhaps some more detailed text from the laws around the time of these changes?
I'll do a little more looking and if I find something, I'll post it.