My recommendation for any use is the Cat-Eye Enduro or its clone, the Cat-Eye Mity. I have these on five of my bikes and my son, son-in-law and a friend have eight more among them. They have ALL been 100% reliable and weatherproof. In fact two of them are on bikes that are specifically used as rain and bad weather bikes.
Cat-Eye uses CR2032 batteries that are cheap, available nearly everywhere and last for years. They also provide all the ride info I've ever needed; speed, total distance, two independently resetable distances (say daily and the entire trip), ride time, average and max speed and a clock. They can be set for two different wheel diameters so you can use the same head on two different bikes.
The Enduro 2 and Mity 3 have been replaced by restyled but otherwise nearly identical models so they are available at closeout prices (less than $20) from Nashbar, etc. These are screaming deals.
My only other experience says avoid Avocet cyclometers. They have proven extremely unreliable over the last few years.