First I'd like to point at one fact and appeal for a logical approach to understand this hadith. The sahabi to whom the Muhammad told "Paradise is beneath her (or your mothers) feet." wanted to go out and fight in jihad, seeking paradise in case that he might be killed (as many quotes of the Quran would evidence).
So our Prophet understood this and told him that looking after your mother, caring for her, respecting her and staying at her side and in charge for her has the same value -by Allah- as if you went out for jihad and died.
Without checking a hadith commentary I would even assume that this young man was his mothers only son, or the one who was in charge for his mother or family. As this was often a reason for our Prophet to reject young men to go out to fight.
On the whole the hadith doesn't say that it is his mother who will let him enter paradise, but his good deeds (if Allah accepted them as such): in this case treating his mother well or behaving like a good son should do.
From the commentary book merqaat al-mafateh sharh meshkat al-masaabeh مرقاة المفاتيح شرح مشكاة المصابيح of Mulla 'Ali al-Qari (as neither as-Sindi nor as-Suyuti in there commentaries on sunan an-Nasa'i added any explanation to this specific hadith) you may find the following explanation:
"beside her feet" means staying beside her to look after her because the Paradise is beside her feet, as she would be a reason for letting you enter it. At-Tibi added "beside her feet" is an allegory to express the highest abidance غاية الخضوع
and the "maximum" of humiliation نهاية التذلل towards your mother (as one of your parents) as asked by Allah in (17:24). He also added maybe our Prophet knew the situation with this young man or with his mother and deduced that it is the best that he stay beside his mother.
There are other narrations of this hadith like this in sunan ibn Majah and an other narrated by at-Tabarni.
Note that statement "Paradise lies beneath the mothers feet" is considered as weak or fabricated.
Take also a look at these fatwas in Arabic:
From islamqa, islamweb 118550 and islamweb 19310.