To replace the battery in an iPod Mini is not too difficult and the video on this page will show you how to do it.  The 1st gen Minis had not-so-great battery play time (8 hours) with a 400 mAhr battery.  Many of the replacement batteries you can find here are higher capacity, so your play time is longer.  The iPod Mini 2nd gen seems to have different components which don't use as much juice, so they lasted much longer right out of the box.  The replacement batteries are the same for both the 1st and 2nd gen.

 

There are some key things which will help make this successful (or could ruin your clickwheel or Main board if you are not careful).

- During the part where you disconnect the cable on the bottom of the Mini, you MUST disconnect the small hard plastic part from the socket it is in, NOT by pulling on the cable itself. If you pull the cable (which is thin and golden colored) you will probably separate the cable from the connector at the end. While this is not fatal, you will have to get a new clickwheel cable. To achieve the separation, look carefully at the connection (you will be looking sort of "under" the ribbon), there will be a small gap between the connector (on the ribbon) and the socket (on the main board). Use a small flat head screw driver in the gap to pry the connector away from the socket.

 

-During the part where you pull the board out of the body: try to push from the bottom as far as you can before you start pulling from the top. This is because at the top, there is a single connector which connects the headphone mini-board (also called a daughter board) to the main board. I think this is the very worst part of this design because if you pull too hard, the headphone connector will separate from the board leaving tiny leads that are virtually impossible to repair (the connector envelopes these leads but is not physically connected to the board). I have now ruined 4 main boards trying to get the headphone connector off (if a headphone channel is broken).

 

-After the battery is in, make sure you have folded the wires inside the profile of the board... if they stick out too much, they will get pinched and potentially shorted when the board goes back into the body.  Pay attention to the final 1/2 inch of insertion to be sure the wires are not going to be pinched.

 

-There is a point during the re-insertion of the board where it will hit the clickwheel and stop.  You will need to coax the board up and down to bend it a very small amount to get past the click wheel.  Don't just jam it and force it because that won't do any good.