The Felling Area, 
Heworth, Follingsby
The Early Years from if

Heworth gets the earliest mention. 
Per History, Topography and Directory of Durham, Whellan, London, 1894 is this
"Heworth Church, St. Mary's, was probably founded by Ceolfrid, abbot of Monkwearmouth and Jarrow, within two years of the consecration of Jarrow Church, A.D. 684. The twenty-three coins of the realm of King Ecgfrid, found in this churchyard in the year 1812, prove this, as they were most likely dedication coins placed under the foundation-stone of the original church; and as King Ecgfrid was killed in 685, this was probably the year in which the foundation-stone was laid".
No, Mr Whellan you fell for a prank. Newcastle University has debunked the coins and the container as a con.

as well as this

In the mid 12th Century is this

Per 3.1.REG.1a   [1154 x 1166 (Accession of Henry II x death of Robert de Chesney, bishop of Lincoln, T.A.M.Bishop: Scriptores Regis p.44 No.169 gives 1155-8)]
Confirmation in free-alms, by H[enry II], king of England to the Prior and monks of St Cuthbert of all lands, tithes, churches and tenures belonging to the Priory of Durham: their lands in Durham and beyond the bridge with a garden, Elvet and the church of that vill...... Heworth, Jarrow with its church and fisheries in the Tyne, the church of St Hild [ South Shields ], Hebburn, Monkton, Heworth and the other Heworth, Follingsby, with all lands and churches owned by them between Tees and Tyne;
note the spelling of Follingsby...I've heard folks bemoan that the better spelled Follonsby has been changed to the current Follingsby. I agree that the 'on' is better than the 'ing' but the 'ing' is the oldest one

So far, the name 'The Felling' doesn't exist

Per here in the mid 13th Century the Heworth area, which includes what is now known as The Felling was a woodland park ie a private hunting area. From that time onwards...when..not yet known...lots of felling of trees took place all over the North East, indeed all over the UK, Europe, The World... So, why did The Felling get that particular name? Did other areas get cleared piecemeal over a long period of time whereas in The Felling area it all happened at once. Well, it didn't really. Joan Hewitt thinks a big bit happened at once when The Bishop arranged for the Prior to have a big "lodge" and grounds in the wooded hunting park. This may have been what people subsequently called "the felling". Before that the only trees felled over the centuries were for routine maintenance or a specific job eg. for the construction of a war ship