Borrowed my brother's car and it has an odd weird starting issue need
 

So this is a 2004 non-S car and at one point in the past it had an ignition interlock installed on it. I am wondering if the removal of that interlock was done in a way that might have caused what I am seeing.

About 1 out of every 5 starts the car just totally freaks out. The dash lights will either start blinking very dim or just the seat belt will come on and still very dim. The other piece of the puzzle is an audible relay clicking and the drivers side window will go down just a little bit more with each relay click.

If I jump start it the car will start. The battery is brand new... literally brand new, it has 2 weeks on it. It was doing this prior to the new battery also, but that is just coincidence, the old battery was due for replacement anyway.

Any suggestions where to start looking? Starter relay? Bad ground? Ignition Switch?? Something else?

Bad ground. Recheck all battery connections

While I don't disagree how does a jump start make it ignore this problem? Is it being grounded by the other car?

So you asked for ideas...described a car that acts like a dead battery yet is running on a brand new one and don't think that some sort of connection is the issue? When you jump the car presumably you are using the connectors under the hood vs. the battery in the boot (trunk) if I assume this is a cooper S. Either way (R52 or R53) minis get very very weird when you have loose ground wires I doubt the ignition interlock system is the issue as it probably interrupted the signal to the relay that gets triggered by the key. My solution is free....who installed the battery? The issue existed before the battery which to me points to a ground or connection issue which might have even misdiagnosed as a bad battery leading you here......

This is a non-S version and the battery is under the hood. I am going to check the grounds, I am just trying to make sense of it since it is not consistent. When it starts (which is most of the time) it is perfectly normal. When it does this freak out it is instantly fixed just by attaching jumper cables.

Oh! I made a mistake too, this is a 2004 not a 2005 model car if that matters at all. (editing the first post to reflect this)

+1 on the bad battery ground. same thing happened to me.

Eventually the clicking/flashing lights were more frequent and it was time for a new battery.

Okie doke, time to start digging for a bad ground... I will report my findings (and then smack my brother in the head for pawning it off on me, LOL)

Look at the ground near the passenger side engine mount....looks like a heavy cable...
Does the car have a sunroof? Any window fogging issues? Just mentioning it because the sunroof drains can become disconnected, and drip water down the front roof pillars...and the corrode the fusebox on the driver side or the bcm on the passenger side....

It appears that it was far simpler than I anticipated. BOTH battery terminals were incredibly loose and pretty badly corroded. I cleaned them up, put them on properly and it seems just fine.

When this happened yesterday it was too dark to see anything useful.

Thanks for the responses!