agrep is definitely a quick and easy base R solution if you have just a bit of data.  If this is just a toy example of a larger data frame, you may be interested in a more durable tool.  In the past month, learning about the Levenshtein distance noted by @PaulHiemstra (also in these different questions) led me to the RecordLinkage package.  The vignettes leave me wanting more examples of the "soft" or fuzzy" matches, particularly across more than 1 field, but the basic answer to your question could be somthing like:
library(RecordLinkage)
col <- data.frame(names1 = c("John Collingson","J Collingson","Dummy Name1","Dummy Name2"))
inputText <- data.frame(names2 = c("J Collingson"))
g1 <- compare.linkage(inputText, col, strcmp = T)
g2 <- epiWeights(g1)
getPairs(g2, min.weight=0.6) 
# id          names2 Weight
# 1  1    J Collingson       
# 2  2    J Collingson  1.000
# 3                          
# 4  1    J Collingson       
# 5  1 John Collingson  0.815
inputText2 <- data.frame(names2 = c("Jon Collinson"))
g1 <- compare.linkage(inputText2, col, strcmp = T)
g2 <- epiWeights(g1)
getPairs(g2, min.weight=0.6)
# id          names2    Weight
# 1  1   Jon Collinson          
# 2  1 John Collingson 0.9644444
# 3                             
# 4  1   Jon Collinson          
# 5  2    J Collingson 0.7924825
Please start with compare.linkage() or compare.dedup()-- RLBigDataLinkage() or RLBigDataDedup() for large data sets.  Hope this helps.