The reason for a delayed probate like this, will be that (a) the family originally saw no reason to get probate (in 1871) but (b) they discovered something, or something happened to change their mind (circa 1911) that required probate. It might be, for instance, that in about 1911, someone discovered a bank account in Ann's name and the bank required probate (or letters of administration actually) to allow the family to have access to the money.
The chances of finding out what that something was are slim, given that the Probate Calendar entry above is probably all there is. However, I suspect the fact that the estate was originally sworn at £5 and then resworn at £265 1s 3d (i.e. the value of the estate was corrected) does in fact tell us something. If it was a bank account that was discovered then, aside from interest, the value would surely be known from the contents of the bank-book, so the initial value would be close to the final. So I don't think it was a bank account that was discovered.
It might have been a share(s) in a company - maybe the nominal value was £5, so probate was got using the £5 figure. Then the share(s) were sold using the probate authority, realising £265-ish, requiring the correction of the estate value.
But one classic possible reason for delayed probate is property. If Ann owned property and her children continued to live there - what would it matter if the house was never handed over legally? It's only when the children, or someone even later, dies that they want to sell the property and discover it's not in their name but in Ann's name, so need probate to administer her estate.
Basically - these are possibilities, no more. There is, however, a way of finding out about property in Scotland. Scottish property transactions were recorded in Registers of Sasines, so you could look in the 1911 sasine registers to see if there is an entry for property owned by Ann (or her executors) being sold. This is not online so has to be done by someone in Scotland - by chance the method of doing this has just changed - see "Sasines records access at NRS now painless" a blog entry by Chris Paton.
I must emphasise that this will only work for property in Scotland - further there is absolutely no guarantee that this is why the late probate was obtained.