I think
@fisicx makes some good points.
And like I said, you can use a colour as part of your brand but it doesn't need to be in the name.
I had some thoughts around a non "random" name, so will share those still.
You make a good point about customers buying me not the kids. My main customers are parents with children who are 3-11 (mainly Mums). However, the kids are the ones who need to remember who/what they saw. They are the ones who come away from the gig with the memories and entertainers and party organisers with a good memorable name do well in that respect. I'm always just James.
One of the biggest problems with a name for me is I don't want it to be too specific. For example I do some magic but don't want to mention that in a name because cool 11yr olds will be alienated (they don't want a babyish party provider). Similarly, the other name that came up in brainstorming was Fresh Parties which is very good for older kids but perhaps not so for little ones.
The kids will likely just say it was a really good show, and the mum will be pleased.
They might say James from company x was really good, but the brand really won't matter to them I think.
The mum will then mention you to another mum, and say James from company x put on a really good show for the kids, arrived on time, communicated well, you should check them out etc.
So the brand really does need to be pitched at the buyer - at the mum.
Personally I don't see what's "uncool" about magic. If I were 11 I'd much rather watch some magic tricks and illusions than the alternative, which would be what, music perhaps? Stand up comedy could be fine.
I don't think the little ones would have any problem with Fresh Parties, they'd just take it more literally, if they even noticed the brand. Not that there's anything "cool" about the name to be honest.
But not wanting to pigeon whole what you do is a good reason to not pick a name with Magic in and so forth, so I agree with the conclusion.
For initial brainstorming you should just throw random ideas around, not rule anything out, use scrabble pieces, word games, and so forth, and be completely free with your ideas.
However for something more structured later on my line of thinking would have been more like this:
Target market - mum's with kids
What do they want - Someone reliable, that can be trusted, will show up in any weather, and put on a decent show - keep the kids entertained.
What you provide - Parties, Entertainment
Possible name - Anyweather Parties
So you're communicating that you provide entertainment regardless of the weather, that you'll show up regardless, solidity and reliability.
The corporate identity could be a cloud, and in place of a sun you'd have a yellow balloon - like a weather pictogram. The cloud could even have a slight magic look.
Key brand colour can then be yellow. With yellow t-shirts etc.
That looks lively, fun, and is pretty neutral, as well as often used by personnel at leisure facilities.
You could then even make a joke about your surname when you introduce yourself as James from Anyweather Parties
It can also work as Anyweather Entertainment
But yes, you might just want to get the scrabble and post it notes out, a big sheet of paper and some pens, and throw some letters and words around.
But if you were to go with the real word route, I would follow some reasoning more like that above. Do start as free as possible though.
Later you can always run a poll, including on here, with a shortlist of names.
I hope that helps.
If you have any questions on the hosting side of things then feel free to get in touch.
Best of luck,
Dan