Just like Eddie.

There are two answers, really, because the first recording of an electric blues guitar was for demonstration purposes only. It was during the days when Rickenbacker was testing out the first electric guitar which guitar geeks (sorry, aficionados) will know was nicknamed the frying pan.
In 1932, a 28-year old white band leader and Hawaiian guitarist called Gage Brewer, from Wichita, Kansas, recorded Eddie Green’s 1918 blues composition, ‘A Good Man is Hard to Find’, as a demonstrator
record for the new electric guitar company.
While not a commercial recording, this is thought to be the first time anyone had played an electric guitar publicly. Brewer later took the new electric guitar on the road with his band; and broadcast the Rickenbacker’s exciting new sound over local radio.

This young trio of virtuoso musicians, featuring Eddie Durham, right, played 12 instruments between them while backing blues diva Mamie Smith on tour in America in the 1920s. Valentine Billingham’s centre and Edgar Battle left.

Probably the first blues-oriented electric guitar ever recorded commercially was by the virtuoso trombonist and guitarist, Eddie Durham. Of African American, Irish, Mohawk and Cherokee Indian descent, Eddie Durham was from San Marcos, a place so deep in the heart of Texas, he spoke only Spanish as a youngster. In September 1935, when Eddie was 29, he recorded some bluesy electric guitar licks with amplification he had rigged up himself, on the swing era tune, ‘Hittin’ The Bottle’, for the black bandleader, Jimmie Lunceford.
Eddie Durham would later teach Charlie Christian how to play the electric guitar and is also credited with recording the world’s first electric guitar jazz solo on Lester Young’s Kansas City Five sessions in 1938. Eddie Durham co-wrote Count Basie’s ‘Topsy’, arranged Glenn Millar’s iconic Second World War anthem, ‘In The Mood’, and worked with everyone from Billie Holiday and Cab Calloway to Artie Shaw and Earl Hines. 
Listen to Eddie Durham in 1935 below. His electric guitar comes in after 48 seconds.