I would like some input on the size and specs of a good speaker choice for the 1Wamp.
4 ohm - 8 ohm - 16 ohm, what would be a good choice for this amp?
Planning on using a cigar box for a cab and velcro to attache the 1Wamp to the top of the cigar box for a mini head and cab.
4" speaker is as big as I think will fit in a cigar box.
If the LM386N4 is outputting 1 Watt, what Wattage rating should the speaker have?
Would a 3 Watt speaker be louder than a 5 or 10 watt speaker? Can the speaker be too high Wattage?

Hi, thats a very interesting question, you can find some nfo/ideas here:
electronics.stackexchange.com/questions/...e-used-with-an-lm386
www.electro-tech-online.com/threads/lm38...-sound-clear.105376/
The parameter that defines how loud is a speaker is the "sensitivity", basically the higher, the better and the louder.
Placing a 100Watt speaker wont make the system louder, it would possibly go in the other direction because usually higher wattage speakers are less sensible.

So, as a rule of thumb find a nice speaker that fits in your enclosure, with a wattage of 1 watt or more, but dont get crazy with the watts.
Also the enclosure size and material make a big impact into the sound.

Also if you have a look into the lm386 datasheet, it can drive anything from 4 to 32 ohms, but recommend 8ohms, so if you are buying new stuff, go for 8ohms (they are usually more available)

Ray wrote:
I think what I'm really after here is what make one speaker better than the other?
$2.00 speaker from adafruit or a $10.00 from Visaton? both 3 inch both 4 ohm adafruit is 3 Watt and Visaton is 5 Watt.
I have a 12" 8 ohm cabinet from Bugera and have run the 1Wamp into that speaker. Same amp on the same setting with the same guitar.
Obviously the 12" Bugera sounds better, but it also has less distortion at the same amp and guitar settings. I believe it is rated at 20 or 25 Watts.
I understand why it is louder, it is moving more air with a 12" cone and it has a sensitivity rating of 98 db.
BTW, I'm in a 1-car garage converted to an extra room and the 1Watt through the 12" will run you out of here, and it sounds really good.
I want the get the best speaker for sound quality that will fit in a cigar box size box.
I'm leaning for 8 ohm and 10 Watts in the 2.5" to 3.5" size. (since my 1Wamp drives the 12" plenty loud)
I think either one of the two speakers below would be a good choice.
www.parts-express.com/fountek-fe87-3-ful...river-8-ohm--299-022
www.parts-express.com/dayton-audio-ce-se...iver-8-ohms--285-143

Thats a very simple but at the same time very complex question. There are so many factors that influence the " sound " of a speaker that it is not easy to find the perfect speaker just reading datasheets.

Usually the frequency response, power and sensitivity are pretty important but also the enclosure, and port (or absence of port) make a great impact into the sound.
I woud try to get a speaker within my budget, with the correct wattage, size, good sensitivity and a decent frequency response (have a look into the Marshall MG Analysis in ElectroSmash - to the speaker section - , I am sure you will get ideas and also you will see how reduced are the frequency response in guitar amps)