"Goes off during the next storm or brownout."

It should.
Shouldn't be on during a lightning storm, and of course will go out during a brownout.

1) A nearby lightning strike can cause a power surge, that may result in;

A) A bad Power On switch
B) A bad Power Supply
C) Go past the Power Supply and fry out the main hardware components.
(To include; Processor, Ram Memory, Harddrive, graphics chipset, etc)

2) Same with a Brownout. A Brownout can also cause a power surge.

{Surge protector? Do you have one that costs around $350 to $400? Otherwise the lessor versions are just that.
Lessor.
They can't protect during a lightning storm, or brownout. The advertising is false.
(Do I use these expensive surge protectors? No}


From your diagnoses of a quick blink of the Power On LED, when the Power On switch is pressed,
the problem could be the Power Supply, MAY be the motherboard.

I do not think there is a cold solder joint.
Isn't logical. The computer would have messed up before now. This would have been noticed when the computer was not too old.


There are electronic components that I feel may be at the crux of this problem.
Electrolytic Capacitors.

Electrolytic Capacitors are the weakest link.

There are Electrolytic Capacitors in the Power Supply, and on the motherboard.
They are used as Filters, and Voltage Regulators.

The type of Power Supply used in your Presario desktop is an SMPS.
Switched-Mode Power Supply,

http://en.wikipedia.org/wiki/Switched-mode_power_supply

Click on the photo at the top right to enlarge.
The two electronic components that have a blue ring, and one is marked with the letter B.
These are Electrolytic Capacitors.
You are looking at a Top View.

Same with the one's that have a black ring, to the right, and surround the letter E.

These are Radial Aluminum Electrolytic Capacitors.
A side view of a Radial Aluminum Electrolytic Capacitor,

http://en.wikipedia.org/wiki/Electrolytic_capacitor

Click on the second photo down on the right. The blue capacitor that has two leads coming out of the same side, (Bottom actually), is an example of a Radial Aluminum Electrolytic Capacitor.
(Has 160V and 10uf on the side. { 160 Volts and 10 Microfarads)

Basic construction of an Electrolytic Capacitor, reveals how it may work in a weakened state.

Using the last example the blue plastic you see is a plastic sleeve.
This goes around the outside of the Case.
The case is essentially an aluminum 'can' with no top, or bottom.

The top is a Vented Cover.
A flat, very thin, round aluminum disk with a shape etched partway into it.
The shape is usually an X or K.

The bottom is a Bung.
A flat, round synthetic rubber disk.

Inside the 'can' case are three strips.
1) A thin aluminum foil metal strip
This is the Conducting strip. The Positive lead is attached to it. (Lead - think wire)

2) A thin aluminum strip with a non-conducting medium applied to it.
This is the Non-Conducting strip. The Negative lead is attached to it.

3) A paper-like strip that is soaked with Electrolytic Paste.

The paper-like strip is laid in-between the two metal strips, and all three are rolled up tightly, then inserted down into the can case.
The two leads protrude down through the synthetic rubber Bung, at the bottom of the capacitor.

When an Electrolytic Capacitor goes bad, the paste inside develops a gas. Hydrogen Gas.
The gas expands, and pushes against the seals. The Vented Cover at the top, and the Bung at the bottom.

The etched shape of the Vented Cover may crack open.
If this has transpired, you may see paste as a yellowish/brownish ooze coming out.

The synthetic rubber Bung may have one side slightly pushed out.
A yellowish/brownish ooze may be coming out. (Capacitor usually sits at a tilt, if this has transpired)

So much paste loss, and the capacitor can operate at a weakened state.
Too much paste loss, and the capacitor fails.

The first Radial Aluminum Electrolytic Capacitors to check are the ones on the motherboard.
A CLOSE visual inspection may reveal bad capacitors,
(Close as in remove the motherboard, and use a bright light, and magnifying glass, may be needed)

http://www.capacitorlab.com/visible-failures/index.htm

There are different capacitors. For the Electrolytic Capacitors used in your personal computer, I will simply refer to them as capacitors.

The capacitors used on the motherboard are also used as Filters, and Voltage Regulators.

Ones that surround the Processor are generally used as Voltage Regulators.
There are other areas the capacitors are mounted on the motherboard, that are also part of the motherboard voltage regulator circuit, but I would like you to view these first.

Part of what the motherboard voltage regulator circuit does, is to regulate voltage for the Processor.
Voltage for the Processor MUST stay within a very specific range.
A Processor has a TIGHT tolerance range.

HP Support > Compaq Presario Media Center SR1830NX Desktop PC > Motherboard Specifications,

http://h10025.www1.hp.com/ewfrf/wc/document?docname=c00496280&tmp_task=prodinfoCategory&cc=us&dlc=en&lang=en&lc=en&product=1843675

Scroll down to the photo of the motherboard.
View the black rectangle, that surrounds the whitish rectangle with multiple holes in it.

{The whitish rectangle with multiple socket holes is the Processor socket. It is a Socket 939 processor socket,

http://en.wikipedia.org/wiki/Socket_939

The black rectangle is a plastic bracket used to hold the Fan/Heatsink combo, to the top of the Processor }

See the silver 'dots' with a black ring around them, and to the left of the Processor socket?
Six of them on the left side?
Three of them close together, then two close together, then one?
These are part of the motherboard voltage regulator circuit, that deals specifically with the Processor.

Observe to see if any of the Top Covers are popped open, and/or the capacitors may be leaning to one side, because one side of the synthetic rubber Bung has been pushed out.
See if there is a Yellowish, or Brownish, or Yellowish/Brownish ooze coming out.

This is a common problem.
Observe the other capacitors to see if there are visual signs of failure.

Do I think the reference of the 'tech' that the motherboard may be bad, is a reference to a bad motherboard chipset? No.
The Northbridge chip, and Southbridge chip usually goes out all at once.
Also everyone seems to want to blame the motherboard, when they can't find the problem.

[The motherboard chipset for your motherboard is the Northbridge and Southbridge chip.

The Northbridge chip is sitting under the aluminum square Heatsink, that looks like a meat tenderizer, and is under the Processor socket.
The Southbridge chip is further down towards the right, under the blue, and black Ram Memory slots, and has ATI on it.

If you see the surface of either one has darkened, or bubbled up, it is bad.

{ The Heatsink for the Northbridge chip has a retaining clip. However, removing the retaining clip alone may not release the Heatsink. There may have been a type of Thermal Paste used, that has glue in it. A hair dryer set to low heat, and moved back, and forth on the Heatsink, will loosen the glue, and allow the Heatsink to be gently twisted, for removal.

ALWAYS thoroughly clean the top of the Northbridge chip, and bottom of the Heatsink, and properly apply fresh, new Thermal Paste. Do not recommend using thermal paste that has glue in it.
Use a scraper such as a clean unused plastic credit card. Then follow with Q-tips dipped in Isopropyl Alcohol. May take a FEW Q-tips.

CAUTION!!
Isopropyl Alcohol is EXTREMELY FLAMMABLE!
Use in a Well ventilated area with NO sparks or flames present }

Do I recommend opening the Power Supply to check it's capacitors?
No sir, I Do Not.
There may be Electrolytic Capacitors that still hold a charge. They can hold a charge for Weeks, Months, sometimes over a year.

IF your fingers touch the terminals (Leads) at the bottom of a capacitor, the charge can be released to YOU.
IF your fingers complete a circuit, that one or more capacitors are in, the charge can be released to YOU.

There is a proper method to discharge capacitors before working on an electronic component.
I will not detail it out here.

The recommended method is to use a Known to be good, Compatible, power supply, for a test.
Perhaps there is one in an unused computer sitting around. Maybe someone has upgraded to a better computer.

Otherwise if you do not see bad capacitors on the motherboard, you can test the voltages of the Power Supply, or use a known to be good compatible unit, for a test.

Continued in a Comment.