How our country came up with name " Nepal " ? Does anyone knows about this ?

A rishimuni named "Ne" had his dwell ("Pal") in the valley. So named the Ne-Pal. Kathmandu was originally Nepal. Well, its etimology also given in wiki.
But did Prithivi Narayan Shaha formally gave the name to united kingdom as Nepal?

Most logical answer is as mentioned by Khairey above. Gyan Mani Nepal also proposed that Nepal is named after Nripa people who lived in Nepal and there are many other propositions from many historians based on what Nepal means in other languages.

Do not know for sure whether Prithivi Narayan Shah changed the name of his country to Nepal from Gorakha, but at Rana Bahadur's Shah's reign it was already called Nepal (Tibet-Nepal treaty, Tibeto-Nepal War). My guess is most likely he did, as he switched capital from Gorakha to valley and embraced many cultures and traditions of valley.

Looking of meaning of nepal in sanskrit or nepali language is wrong. We have to look on what is origin of nepal. In old days, only kathmandu valley is referred as nepal. Even now, older people says they are going to Nepal which refer to kathmandu. So nepal is derived from name of kathmandu valley before Prithivi Narayan Shah came here. In Newari, Nepal is called Nepa de (de means desh or country). The language and the people are called Newa (wa mean to speak). So the real root is Nepa not Nepal. So we should try to find meaning of Nepa instead of Nepal. Newars refer kathmandu is nye, lalitpur as yala and bhaktapur as khopa. The valley was name nye-pa which means flats of kathmandu.

Also Newar's use a different calender with tithis which is based on lunar calender. The new year is on the day of Govardan puja (day after laxmi puja). The calendar is called nepa sambat (nepal sambat in nepali) and it is running in 1130 years now.

Looking the meaning of Nepal in Nepali or Sanskrit is like searching for origin of kumari in Gorkha.

malai ta tha bhayena mathi ka mitra haru le bhannu bhayeko sahi cha hola hai ???

Nai - Naitikta ko niyam, dharma ko niyam
Nai ka palan garney lai Nepali bhaninchha
Nai ko baran/dharan garney lai Newari bhaninchha

Nepal - Nepali ko desh, Newari ko desh, dharma ko desh
(by SN Goenka on May 21, 2005 at Nepal Vipassana Center)

I can't believe no one has posted the meaning of Nepal that we learnt when we were kids.

NEPAL stands for Never Ending Peace And Love!

But then, I guess this is not true any more....

Oh, and JAPAN was Jumping And Pumping At Night!

well, the name nepala is given as one of the 16 mahajanapadas at the time of buddha.so i personally dont believe on its origin being on newari  term 'Nepa'.newari language was spoken later which is a mixture of prakit elements  in tibeto-burmese language.it is possible that nepala became nepa during the malla period and people started giving its own meaning.