Make a spreadsheet! and start weighing parts.
The frame might have the biggest weight to lose, but on a £/gram basis they almost always come out as a bad place to start. to get better he'd either end up with an aluminium frame (as good carbon costs too much for the weigh saving) or chinese imports.
Wheels are always a good place to spend money (unless you ride into potholse or crash regulalry). Novatech hubs, DT revolution spokes and Stans alpha rims would weigh <1300g and cost about £250, tubeless weighs slightly more than ultremo's and conti supersonics but both circa 230-260g each end. For a bit more money/bit less useability, planet-x/chinese carbon ~25mm tub rims weigh almost nothing (well, 300g for a rim), but are carbon (poorer wet braking) and tubs.
After that I changed my chainset to a 2nd hand FSA carbon one, the groupset to 2nd hand DA, £110 and £180 respectively, and the groupset especialy shed a lot of weight compared to the tigra/105.
Planet-X brakes were a nice upgrade, significanlty cheeper and lighter than DA, power is OK rather than outstanding.
Finishing kit shed some weight, but I went for stuff I liked rather than light. Ended up being light-ish, but could have gone lighter.
It makes the bike feel completely different, but not actualy that much faster. It probably is faster, but that's probably the tyres and rims construction/profile rather than weight.