Smth along these lines should do it:
names = c("Adams","Adams","Adams","Adams","Bobby","Bobby", "Charlie")
years = c(2002,2003,2004,2005,2004,2005,2010)
Run_value = c(10,15,15,20,10,5,5)
library(data.table)
dt = data.table(names, years, Run_value)
dt[, lag1 := c(NA, Run_value), by = names]
# names years Run_value lag1
#1: Adams 2002 10 NA
#2: Adams 2003 15 10
#3: Adams 2004 15 15
#4: Adams 2005 20 15
#5: Bobby 2004 10 NA
#6: Bobby 2005 5 10
#7: Charlie 2010 5 NA