From a branding POV I'd be advising a start-up that initially it would be good for the brand name to clearly reflect its an ice cream business. You want people to instantly know that so if you're at events / on shelf you'll get immediate footfall / eyeballs due to the clarity. There will be a drop off if people don't instantly know what you're selling as an unrecognised brand.
You need to test and prove the business model and branding before thinking about branching out unless you have a five year plan and solid investment. It's just too risky otherwise.
I would personally register a company under an umbrella name that allows you to branch out in future but use a public brand name that is more specific to start with.
Then you have the option to continue functioning seamlessly behind the scenes if the business expands to different products, but change one word / branding elements to reflect other products.
Think innocent drinks, that's the company but it includes innocent smoothies and now innocent juices. Registering under an umbrella company allows for innocent fruit pots, innocent snacks etc. Clear to customers and building on the innocent brand awareness.
Think about your market - do you want to target people who are always looking for organic food choices "finally an organic ice cream" and therefore will be won over on this element, or do you want people who will be won over on flavour "and it's organic too so even better". Never ever say your market is "everyone".
For a start up it's much simpler to have a range that caters to a niche, because you can do really targeted branding and marketing.
For example if you focus on the organic food market you are better of going to three smaller organic festivals with a captive audience than going to a larger festival where there will be lots of people who would (for example) choose the cheapest purchase because the organic element isn't important to them.
I would steer clear of branding that is a really cheesy pun or has a z at the end to try to make it sound "fun" as this dates badly and cheapens the brand.
And be sure to test flavours and branding on people who aren't just friends and family. For example samples for feedback works well - mini portions to the public at events in return for their honest thoughts. If they don't like it you've learned valuable lessons and if they do they are likely to purchase the full size version there and then, so if the product is strong enough you cover the cost of the samples with the profit from the full size purchases. And if it isn't strong enough you have loads of feedback to work with and won't be flogging a dead horse. The vast majority friends and family will always say everything is great so don't rely on them!
Do lots of research on competitor brands so you don't unknowingly end up using part or all of a brand name that already exists - you can search on companies house for company names so do this for all elements of any brand name you're considering.
Hope some of this helps 