All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
Every Muslim is encouraged to perform the Tahajjud prayer, which is a recommended Sunnah, regardless of their sex, age, or social status.
Indeed, Allah praised the people who perform the prayer by night in His Book and in the Sunnah of His Messenger and He promised them a great reward; Allah says (what means): {They arise from [their] beds; they supplicate their Lord in fear and aspiration, and from what We have provided them, they spend. And no soul knows what has been hidden for them of comfort for eyes as reward for what they used to do.} [Quran 32:16-17]
Also, Allah informed us regarding the elite of His slaves that they deserved Paradise because among their qualities is that: {They used to sleep but little of the night. And in the hours before dawn they would ask forgiveness.} [Quran 51:17-18]
Moreover, the Prophet said, “The best prayer after the obligatory prayer is the one performed in the deep of the night.” [Muslim]
Therefore, it is permissible for the unmarried girl to perform the Tahajjud prayer, just like married women, and what that person told you is not true.
Allah knows best.