The 99 Avenger I just bought with a 2.5L had sat for about 3 months when I bought it. Started it and drove it home with no noises. The next morning same thing, started quiet, everything normal. Did an oil change and the following morning (yesterday) when I started it there was a pretty noticeable tick or rattle that seemed to be up top. I shut it off and installed my new stereo then restarted it and the noise seems to be gone or at least so faint its not really noticeable.
The car has 145000 km's on it and seems to have been well maintained. I did have an issue finding an oil filter that would fit between the o2 and the threaded part of the adapter. It had a Fram on it which cleared easily. I have my reasons for not wanting a fram on my vehicles so I bought a wix filter that was supposed to be correct, it was slightly longer and the gasket suface was raised a bit more, also the holes between the threaded one and the gasket were raised as opposed to dipped down on the fram, also the bottom of the filter had more of a dome on the fram allowing more clearance around the o2. Tried an Auto Xtra (made by wix) same issue.
Bought a Purolator which I have never tried, It was snug but fit. Is it possible its a filter issue causing my start up noise?
Im using 10w30 Valvoline oil which is not normally my first choice but I have never had issues before with it.

I also noticed my oil psi seems a tad lower at idle but if i just touch the gas once for a split second it seems to come up and remain there at idle. Didnt seem to do that before the oil and filter change.
Sorry for the long post but thats all the info I could think of.

 

What oil did you use ? sounds like a lower viscosity is lowering the pressure and amount of oil to head causing the noise. With that many miles the oil pump could be weak/sludged up passages as well

 

I used Valvoline 10w30.
Its only 145k kilometers not miles. Would there really be oil pump issues already?

 

Since you don't know the history of the car and oil changes/type of oil I'd recommend an oil flush to see if that cleans the passages out /sludge out and fixes the noise

 

I would go with 5w30 as recommended by Dodge. The thicker oil may be taking longer to reach the overhead camshafts and components.
I also would recommend a Mopar filter and 5w30 Pennzoil. That's what my dealer uses (except 5w20 on most newer vehicles)
I too, will never use a cheaply made Fram oil filter. Their air and fuel filters are good, but not the oil filters.