Use scoreboard operations
I assume you also have the time stored in a non-formatted scoreboard, e.g. in ticks. Let's call that objective time. If not, see below.
Make another dummy scoreboard objective called best, and initalize it to 0, e.g. by running
/scoreboard players add @a best 0
When a player finishes the race, mark him in some way, e.g. by assigning a tag (I'll use tag=finished). Then simply run
/execute @a[tag=finished] ~ ~ ~ scoreboard players operation @a[c=1] best < @a[c=1] time
The < operator will assign the lesser score of best and time to best. The execute makes sure that both targets of the scoreboard command are the same player.
You can turn best into bestminute and bestsecond the same way you converted time into the display stats.
Converting times to a single number
If you don't have time stored as a single continuously increasing number, you can use scoreboard players operation commands to turn your dispminute and dispseconds scores into one. Set up the new objective, and store the number 60 in a fake player (called #SIXTY):
/scoreboard objectives add time dummy
/scoreboard players set #SIXTY time 60
Now, set up a repeat/chain line and run.
/scoreboard players set @a time 0
/execute @a ~ ~ ~ /scoreboard players operation @a[c=1] time += @a[c=1] dispminutes
/execute @a ~ ~ ~ /scoreboard players operation @a[c=1] time *= #SIXTY time
/execute @a ~ ~ ~ /scoreboard players operation @a[c=1] time += @a[c=1] dispsecond
This will first reset the time score for every player. Then, the players' scores in dispminutes will be added to time, and multiplied by 60. Lastly, dispseconds is added to time.
