Grammaring [reformatted] gives a fairly comprehensive list of nouns which may take a to-infinitive complement:
Nouns [which may be] followed by the to-infinitive:
ability // advice // agreement // ambition // anxiety // appeal //
arrangement // attempt // chance // choice // decision // demand //
desire // determination // dream // eagerness // failure // goal //
intention // motivation // need // offer // opportunity // order //
permission // plan // preparation // promise // proposal //
recommendation // refusal // reluctance // reminder // request //
requirement // suggestion // tendency // way // willingness // wish
- The ability to cooperate with others is as important as managing on our own.
- Our decision to close the firm was a difficult one to make.
- We were surprised at his offer to take us home.
- As a result of his failure to pay the mortgage, his house was foreclosed.
- She showed no willingness to help.
It will be seen that 'idea' isn't included in their list.
Neither does MyEnglishTeacher include 'idea' as one of the '37 most common noun + to-infinitive [colligations]' (a virtually identical list).
However, 'It was X's idea / her idea / ... to buy a new door' etc are unarguably idiomatic and totally acceptable.
Longman's advice is:
Don’t say ‘the idea to do something’. Say 'the idea of doing
However, you can say 'it is a good idea to do something' and 'it was
someone’s idea to do something'.
I'm indebted to Phil Sweet who refines Longman's crude rule of thumb in what I consider to be a very reasonable way:
I use the to-infinitive for stuff that actually happened, and the of + ing
form for stuff that hasn't (as yet).
- The idea to finish the basement and rent it out [was John's] .... [implies that] this was done.
- The idea of finishing the basement and renting it out [seems reasonable.] ... is [more] speculative or theoretical in nature ...
it's something being considered.