Yes, the ability will resolve and grant +1/+1 to the 1/1.
The condition on Consul's Lieutenant's second ability ("if it's renowned") is an "intervening if" clause.
603.4 A triggered ability may read “When/Whenever/At [trigger event], if [condition], [effect].” When the trigger event occurs, the ability checks whether the stated condition is true. The ability triggers only if it is; otherwise it does nothing. If the ability triggers, it checks the stated condition again as it resolves. If the condition isn’t true at that time, the ability is removed from the stack and does nothing. Note that this mirrors the check for legal targets. This rule is referred to as the “intervening ‘if’ clause” rule.
As the bolded part indicates, the Lieutenant's ability only triggers if the Lieutenant is renowned when it attacks, and the ability only resolves if the Lieutenant is still renowned when the ability resolves.
So when the ability is about the resolve, it asks "Is my source renowned?" But in our scenario, the Lieutenant doesn't exist at this time. So how is this question answered? It's answered using Last Known Information.
112.7a Once activated or triggered, an ability exists on the stack independently of its source. Destruction or removal of the source after that time won’t affect the ability. Note that some abilities cause a source to do something (for example, “Prodigal Pyromancer deals 1 damage to target creature or player”) rather than the ability doing anything directly. In these cases, any activated or triggered ability that references information about the source because the effect needs to be divided checks that information when the ability is put onto the stack. Otherwise, it will check that information when it resolves. In both instances, if the source is no longer in the zone it’s expected to be in at that time, its last known information is used. The source can still perform the action even though it no longer exists.
When it ceased to exist on the battlefield, the Lieutenant was renowned, so the ability will resolve.