This is a copy of the reply I received from the Southampton Local Studies and Archives,
1st Endle/River St
In the 1899 edition of Kelly's street directory, the street's name is still River Street, changing to Endle Street in the next edition of 1900. In The Hampshire Advertiser of 26 August 1899, there is a death notice for someone living in Endle Street and the issue of 26 August 1889 has an advert for a block long-leaseholds for 53, 55, 57 and 59 Endle Street (late River Street). This newspaper is published online on the British Library database of nineteenth century newspapers.
The following entry is from the 11 October 1898 minutes of the Works Committee of the County Borough of Southampton -
River Street
Also resolved, - that upon the receipt of a Requisition from Ratepayers and Owners of property in River Street, the Committee will be prepared to consider the subject of the alteration of the name of the Street, a suitable name being submitted for that purpose.
The minutes do not record the suitable name but I presume that it was a suggestion from Messrs F and J Young who, according to the same report, had enquired if River Street had been taken over by the council and suggested an alteration in the name of the street (again, no record of the suggested name). F and J Young was a firm of builders and decorators in St Mary's Road. A search of the council minutes for the following years did not produce anymore information on River/Endle Street.
2nd is a report from the Hampshire Advertiser 28/10/1899, where both River St and Endle St is used, between members of the same family!!
there are quite a number of reports of various kinds I found it all so interesting,
bowman