These can be simply put as:
Niyaz of The Be-Niyaz
The Niyaz who is be-niyaz
In the first case the word niyaz is used in the meaning of nazr (offering ) and Be-Niyaz is used for Allah who has the quality of being be-niyaz (who is unaffected) on each and every thing. So calling Hazrat Qibla (R.A.), Niyaz Be-Niyaz means that he is an offering to Allah,the Be-niyaz from his parents,his Murshid or by himself.This is justified by his devotion towards Allah.
This is similar to saying niyaz of Hazrat Gaus Paak (R.A.) which indicates to any particular item which is offered in the name of Hazrat Gaus Paak (R.A.).
The word niyaz also used in the meaning of haajat (requirement) or khwaish.So using this meaning ‘Niyaz Be-Niyaz’ indicates Hazrat Qibla Sarkaar of being a person who has the khwaish of Allah or who is in search of Allah.
In the second case Hazrat Qibla Sarkaar is said to be ‘be-niyaz’, who is be-parwaah.The one who is unaffected by something or who doesn’t requires something or who is not inclined towards something. Sometime people object saying that be-niyazi is a quality of Allah and this quality shouldn’t be associated with anyone.It has to be understand that Allah is be-niyaz on each and everything but when Hazrat Qibla is called be-niyaz he is said to be so on certain things, and the absolute authority for this rests with Allah . As Hazrat Qibla sarkaar has himself said :
Yahaan tak dia mujhko husn-e-uruuj
Ke bande se maula banaya mujhe
The basic concept of taswwuf is to leave the love of world for Allah and the life of Hazrat Qibla Sarkaar was the perfect example of this and so he is not only be-niyaz but also can be called be-niyaz-al-khlak ( one who is unaffected or uninclined to all worldly things ).
In the the relationship between aashiq (lover) and maashooq (the one who is loved ) the maashooq is sometime is said to be irresponsive towards the action of aashiq.For this poets have used the word be-niyaz.
As Mirza Ghalib says:
Hum bhi tasleem ki khoo dalenge
Benayazi teri aadat hi sahi
Beniyazi had se guzri banda parwar kab talak
Hum kahenge hal-e-dil aur aap farmaenge kya
Zer e shan e beniyazi khul gai
Jis taraf rukh ko phiraya yar ne