Quran says in 2:197 "Hajj is [during] well-known months". Why Quran uses "months" (أشهر) when hajj is presently done only during Dhu al-Hijjah month ?
Well the Mufassirin say: That the Months of Hajj are: Shawwal, dhul-Qi'da and (at least) the first ten days of dhul-Hijja because if you put on ihram in one of those months you are considered to have put on ihram for Hajj and anything which is prohibited for a Hajj is prohibited for you from that moment!
This Ayah indicates that Ihram for Hajj only occurs during the months of Hajj. This was reported from Ibn 'Abbas, Jabir, 'Ata', Tawus and Mujahid. The proof for this is Allah's statement that Hajj occurs during known, specific months, which indicates that Hajj is not allowed before that, just as the prayer has a fixed time (before which one's prayer is not accepted).
Ash-Shafi'i recorded that Ibn 'Abbas said, "No person should assume Ihram for Hajj before the months of the Hajj, for Allah said:
الْحَجُّ أَشْهُرٌ مَّعْلُومَـتٌ
(The Hajj is (in) the well-known months.) (tafsir ibn Kathir)
Note that you could intent to perfrom a hajj in the form of tamatu' التمتع (See What is the difference between Hajj Mofrad, Mokren, and Motamatte3?) which means performing a 'umrah first then taking off your Ihram and then perform the hajj which is clearly refered to in the verse before (some say that the statement later in the same verse saying: "This is for those whose family is not in the area of al-Masjid al-Haram." also is a reference for qiraan as an option for those who come from far away):
... And when you are secure, then whoever performs 'umrah [during the Hajj months] followed by Hajj [offers] what can be obtained with ease of sacrificial animals. ... (2:196)
you are allowed to start doing this from the month of Shawwal on (note that this form is the most recommended and best according the hanbali madhhab). Also if you performed a hajj in the form of ifraad الافراد you may perform hajj take off your ihram (this form is recommended and considered as the best by the maliki's and shafi'ia) and then intent to do 'umrah in dhul-Hijja.
An other possibility is that Arabs when saying "i visited you on Thursday" mean only for some time on this day, so a time could be expressed by a few of it or a lot of it in this case 2 months and about 10 days until (3 months) stand for a couple of months! To explain and understand this will need deeper understanding of Arabic language specifications!
I just had this inspiration, which is hopefully helpful and a comment on the first paragraph of my answer. We know that an important part of worship and any deed is the intention (an-niyyah). Scholars have defined the intention as what we may call intention with meanwhile start of doing this deed or worship. So we can only make the intention for hajj when we put on Ihram or at least do our first step out of our home towards Mekkah.
So if you said now in rabi'i at-thani I intend to do hajj this year, this isn't a valid niyya, but something scholars would call qasd (you plan to do it, but you didn't start it as well). So only after finishing Ramadan you may be allowed to make the intention for this deed (note that in ahadith the 'omrah of Ramadan is set equal to hajj, which may only refer to the rewards and of course isn't a valid excuse to not perform hajj as well).
So let's assume you are doing 'omrah in Ramadan and afterwards decided to stay and perform hajj in that case you might do your intention at a time in shawwal. Any other person might have the intention at the end of shawwal or later in dhul-Qi'dah as he/she might have a long way to travel (think of caravans, people going by feet etc.).
Note that even now many people who perform hajj arrive 1 month or at least 2-3 weeks before in Mekkah and wait for the few days of "Hajj" so maybe the main reason is that in early days it was not easy to get to Mekkah and it was a long way so at least once you made your intention for hajj you are in Allahs hands, if he toke away your soul during your travel you might be rewarded. So maybe that's why to intent doing hajj you have a limited time of a couple of months were you may do it therefore the plural form. On this basis one could also discuss why Ramadan was not quoted as part of these months!