Of course I've no idea what the name really was, but several discographies give the name as 'Spotlite Lounge', for instance Peter Losin's Miles Davis Discography:
http://www.plosin.com/milesahead/Sessions.aspx?s=581101
EDIT: There seems to have been a club run by Clark Monroe at 52nd street in New York in the 40s called 'Spotlight Club'.
Daniel, I'd seen a few 'Lounge' references as well which got me to wondering? Shows up on this Dizzy disc page (see 1946).....as well as the Cannonball page (guess Miles played there a few times in '58). Maybe they changed the name in the 60s? No matter...just curious.
http://www.jazzdisco.org/dizzy-gillespie/discography/session-index/
http://www.jazzdisco.org/cannonball-adderley/discography/session-index/
p.s. -- wonder if that Miles material was recorded and released? I'm no Miles expert.
EDIT - Looks as though some of the Miles show was in fact broadcast.
Miles Davis Sextet
Miles Davis (tp) Cannonball Adderley (as) John Coltrane (ts) Bill Evans (p) Paul Chambers (b) Jimmy Cobb (d)
radio broadcast, "Bandstand USA", "Spotlight Lounge", Washington, DC, August 9 or June 30, 1958
Walkin' Jazz Music Yesterday (It) JMY ME 6402
All Of You -
'Round About Midnight -
* Miles Davis - Four-Play (Jazz Music Yesterday (It) JMY ME 6402)