Answer: You have raised a very good question. I have taken that for granted all this while too, so it provoked me to look for the answer.
The Permanent Committee for Research and Fatwa was asked about how to fill the night of Lailatul-qadr, their answer included to increase recitation of the Quran. Islam-QA when asked the same question simply quoted the same fatwa from the permanent Committee. However, no specific dalil was quoted.
Dr. Abdullah al-Faqeeh was asked a similar question, and he also recommended to increase recitation of the Quran AND other good deeds, but again no specific dalil mentioned. The general dalil used were the same dalils mentioned in the Q&A below, and I quote:
This is based on the following Hadith narrated by `A'ishah (may Allah be pleased with her): "Upon entering into the last ten (of Ramadan), the Prophet (peace and blessings be upon him) would tighten his Izar (i.e. he stayed away from his wives in order to have more time for worship), spend the whole night awake (in Prayer), and wake up his family." (Reported by Al-Bukhari and Muslim) She also said: "Allah's Messenger (peace and blessings be upon him) used to exert more efforts (in worship) on the last ten than on other nights." (Reported by Muslim)
Shaikh Abu Bakr al-Jazaa'iri mentioned in his tafsir, Aysar at-Tafasir, on Surah al-Qadr: "6 - Recommended to increase recitation of Quran and listening to it in Lailatul-Qadr because Jibril read the Quran to the Messenger ((SAW) twice in Ramadan." This is good fiqh from the Shaikh, jazahullahu khairan.
Generally, the scholars of tafsir when talking about the verse "Lailatul-Qadr is better than 1000 months," this means that good deeds (in general) during that night will equal the actions done for 1000 months. An-Nawawi said after mentioning this hadith of Aaishah on the doa for Lailatul-Qadr: "It is recommended to increase in making this supplication in that night and it is recommended to recite al-Quran and other recommended zikr and supplications in noble circumstances..."
[al-Azkar- Chapter of what supplication to do when in Lailatul-Qadr]
And, al-hamdulillah, yesterday night in between qiyam, Shaikh Ayman ad-Daqqad gave a short reminder on Lailatul-Qadr since 27th Night is the most hopeful to be Lailatul-Qadr. He mentioned the hadith of `A'ishah (may Allah be pleased with her) who reported that she asked the Prophet (peace and blessings be upon him) "O Messenger of Allah! If I knew which night is Laylatul-Qadr, what should I say during it?" And he instructed her to say: "Allahumma innaka `afuwwun tuhibbul `afwa fa`fu `annee (O Allah! You are Oft-Forgiving, and you love forgiveness. So forgive me)." (Reported by Ahmad, Ibn Majah and At-Tirmithi)
He said regarding it that a good fiqh of the hadith is that making this doa is more recommended than reciting the Quran and other acts (apart from Qiyam), since the prophet advised her to say make the supplication and that Aaishah (RA) would only make this supplication during Lailatul-Qadr.
ويجتهد فيها بأنواع العبادات من صلاة وغيرها وقال سفيان الثوري الدعاء في تلك الليلة أحب من الصلاة ثم أفاد أنه إذا قرأ أو دعا كان حسناً وكان صلى الله عليه وسلم يجتهد في ليالي شهر رمضان ويقرأ فيها قراءة مرتلة لا يمر بآية رحمة إلا سأل ولا بآية عذاب إلا تعوذ ذكر ابن رجب أن الأكمل الجمع بين الصلاة والقراءة والدعاء والتفكر وقد كان عليه الصلاة والسلام يفعل ذلك كله لا سيما في العشر الأواخر ويحصل قيامها على ما قال البعض بصلاة التراويح
Al-Alusi said: (The prophet) used to exert himself in it (Lailatul-Qadr) with many types of worships: solat and others. And Sufyan ath-Thawri said: Supplcation in that night is more beloved than solat. And he then said that if he recites or supplicates it is good. And (the prophet) used to exert himself in the nights of the month of Ramadan and recite in them in tartil, and when he passes by a verse on rahmah, he would ask for it, and when he passes a verse of punishment, he would seek refuge from it. Ibn Rajab mentioned: The most complete is to combine between solat, recitation (of Quran), supplication and tafakkur (comtemplation), and the prophet used to do all this especially in the last ten nights and the qiyam of it (these nights) by what some people call: Solat Tarawih.
[Tafsir Ruh al-Ma'ani by Imam al-Alusi] Allahu a'lam.