Could someone explain tenant right please

[Claire64]

Will of 1781; an ancestor left his sons, "my tenant-right of the lands I hold under Godfrey Bosville Esquire called Lady-Shaws". 
An inventory values this at £57.  His total worth was £59.5s.6d.
He left 4 of his grandchildren £1 each, this £4 to be paid out of the Lady-Shaws.

Lady-Shaws was near Salters Brook off what is now the Woodhead Pass to Manchester from Penistone, and quite a few miles from where he actually lived.  From what I can make out, this area was mainly used for sheep.  It's very bleak.  Could someone please explain what all this means?

[Marmalady]

Farming tenancies would often be for a number of lifetimes rather than a set number of years -- so that farmers would know the farm would pass to their sons, grandsons etc -- and so would make long term plans for the land.

Maybe he had taken a tenancy on land suitable for sheep grazing whilst living elsewhere on a more congenial farm.

[Maiden Stone]

Your ancestor or someone before him must have entered into a lease agreement with Godfrey Bosville. The lease may have survived in an archive. Find out who Godfrey Bosville was and what he owned. If he owned a lot of land there may be a collection of estate papers.
This was the era of enclosures of common land. Some tenants and commoners had rights to graze animals on common land. They might be limited by number of animals or a set number of days p.a. or at certain seasons.
Another use of the land may have been for cutting peat for fuel.
Terms to look up include 'lease for 3 lives' ,  'lease and release',  'surrender of lease', 'copyhold tenants' . Owner charged a tenant an entry fine to take out a lease for 3 lives. Tenant also paid rent.  £57 might be what was paid on entry or the value of the remaining term if the lease was relinquished. If the rest of his estate was worth only £2 he can't have had much, unless he had debts.

[Claire64]

Thank you for the replies.  I am looking into what survives at the archives, which is looking promising.

He wants the money for the grandchildren to be paid out of the Lady-Shaws land on which he held the tenant-right.  This implies that they derived some income from it. Or could they surrender it and get some money from it?  He didn't live there. 

Any ideas?

[Maiden Stone]

There's a good explanatory guide about tenancies on University of Nottingham library website. I'd bookmarked it but can't find it now.
I've seen documents about lease surrenders in Lancashire Archives catalogue LANCAT. Those I was interested in were Clifton of Lytham estate papers. An owner might offer to buy back a lease to bring a property under his control again or to get rid of a particular tenant.