I recently got a 2002 standard. I noticed once on a while while I'm idling the temperature needle moves up even making the light go on once, but then as I start to accelerate, it falls back down to the proper temp. I checked the reservoir and it seems fine, I don't see any leaks or smells.

 

Sounds like your radiator cooling fan is dead.
As I understand it, the needle doesn't even move over quite a large temp range. The red light is BAD - you don't want to go there.
A new fan assembly can be had for less than $90 (plus install, if you don't do your own work.)

 

+1 definitely fan, but check the fuse first! should be a 5 amp fuse under the hood, don't remember which number but it's definitely a 5A, if it's blown, replace it and if it blows again, then i would replace the fan.

do the install yourself, totally worth the $$$ left in your pocket btw

 

Note: In the early cars, the rad fan is on the same circuit as the power steering fan. If the fuse is blown, check the PS fan as well. At some point MINI offered a retrofit to split the circuit - not sure if they are still offering that.

 

On a justa....(base non-s slang), check the fan....
like mentioned, the ps fan can jam, and blow the fuse for the relay that activates tye radiator cooling fan....pretty common on older cars....in 2005 mini changed the wiring, and modified some early cars under warrenty .....basicly it put the ps fan supply power on the same fuse as the optional backup sensor....
A new ps fan runs about $180 if you reuse the mounting bracket, about 200$ with a new one....simple swap with 2 bolts and one plug.....can be don on a ramp....much cheaper than a blown headgasket from overheating or a failed ps pump $600+ labour....the fan is there for a reason.
grit/sand/water/even plastic bags jam the fan or destory the bearings....

 

Beyond that....
Has the coolant ever been changed....the mini uses non longlife coolant and it the most overlooked mx item on the car....the waterpump could be severly worn, etc, but the ps fan failure causeing the rad failure is mosr common.

 

The fuses are fine and the fan goes on. I also discovered that when the temp. needle moves up, the heat cuts out. I've bleed the lines and that works for a bit. Anyone have any other ideas of what it could be? Someone said that it might be the start of a head gasket going? I hope not.

 

High temp and loss of heat would make me think air in the system.
I think you are on the right track with the bleeding. As the temp goes up and steam forms you would lose heat.
Try elevating the front of the car and get it up to temp before you start to burp the system. Elevation to cause the air to go to the bleed screw on the upper hose.
Good luck.

 

Try to flush the heater core yet? Then bleed it....it can be a pain to bleed a mini...BE SURE TO USE BOTH bleed locations...many try to use only the one next to the radiator....

Listen to your car when you start it up some AM (or some other silent location...no traffic etc when cold)...no fan no radio, windows closed....
Do you hear a "mini waterfall"...a very slight gurgle...if you do that is the sound of air trapped in the heater core.....