Step 1, verify the it is the remote. Use a digital camera, or your phone or camcorder, to check. Simply point the remote at your camera, push some buttons and look for activity on the camera display, it should look like a blinking LED. If you see something, it’s not the remote.
Step 2, open the remote. They’re all different, so you just have to be clever here. After removing any visible screws, sometimes they’re hidden under stickers, work the two halves apart. Sometimes a little twisting is needed to find the seams.
Step 3, if you are not able to solder, go to step 4. Re-flow all the solder joints with a fine tip soldering iron. Make sure you hold together with tweezers anything that might pop out while applying heat. When the solder flows, you’re done, too much heat will damage the board. Also be very careful not to cause any shorts or leave any solder blobs.
Step 4, clean the circuit board with isopropyl alcohol and a clean tissue. Make sure the circuit board contacts, black contacts stamped on to the board, are clean too. Then with another clean tissue and alcohol, clean the button contacts on the rubber key mat.
Re-assemble and check with the camera again. If this didn’t work, time to buy a new one, not much more you can reasonably do.