The best thing to do when choosing a name for a business is to do keyword research using the Google KeyWord tool and see what people are searching for under the keyword "handyman". Then because you are wanting people to find you in a local market you'd add "YourTown Handyman" to the KW search. Choose a name that has the most local and global searches for an exact match. Then when people are searching you have a better chance of them finding you. Then get your business listed on Google Places.
Also go for the .com domain if at all possible. Next best is .org or .net.