should I present solution ideas in a cover letter?
The big danger here would be drawing disproportionate focus on your suggestion, which you are likely giving without an intimate understanding of the research and problems that the people in the research group are privy to. 

Because of this lack of knowledge of the fine details of the work, it can be dangerous to start giving suggestions as it can very easily come off as uninformed (especially to them, as they've been working on it for a while), arrogant/delusional ("does this person really think they've solved the problem after reading a few papers?"), insulting ("of course we tried technique X, does this person think we are morons?"), or woefully misinformed ("that technique is completely inappropriate to our topic - this person doesn't have any idea what we are doing but is trying to pretend they do"), etc.

It could also be a really brilliant insight, and they'll be impressed. Wouldn't that be great?

But realistically, this is like trying to throw a dart across a room at a tiny bull's eye target that happens to be surrounded by people...what's the chance of you hitting the target, vs landing a dart right in someone's backside?

So while this is not something you strictly absolutely must not do, I would urge you to be extremely cautious as it is so very easy to do this wrong. You are interested in their research and have read papers - that's great! Have interesting questions about how certain potential methods might suit their research? Great! Tell them they should just use technique X? Whether you are right or wrong, you run a very real risk of being perceived negatively.