'Can I boil my water the night before brew day and leave in the HLT with lid on ready for the morning? '
Of course you can. This will help reduce chlorine and other volatile substances, precipitate calcium carbonate if you have excess bicarbonate, or temporary, hardness, and sanitize the water by killing microbes.
'Will this have any adverse effect on the water?'
Adverse is a relative term, depending on your goals. If you want to get rid of excess temporary hardness, chlorine, microbes and any other volatile substances, then no, there really aren't adverse effects.
If you were making a darker beer though (where the darker specialty grains would serve to lower the mash pH), the boiling could remove some of the water's ability to counteract this pH change (its hardness) and you might end up with too low a pH. This is why brewing dark beers is particularly well suited to water supplies with high levels of bicarbonate hardness. However this is a concern only if you have high temporary hardness. Boiling has no effect on permanent hardness in water.
'...heating it twice may remove some oxygen, is that a problem?'
No. Strictly speaking, oxygen is even bad on the hot-side of the process (though it's probably not quite as adverse as oxygen on the cold side). Right before pitching yeast is really the only time oxygen should be added to beer.