。
She had a distinctive way of speaking.
Literal
She [topic-は] distinctive way-of-speaking [object-を] was-doing.
話し方 ('way of speaking') is a model example of the productive ~方 suffix: take a verb stem, add 方 (read -kata), and you get the noun for 'manner of doing it.' 使い方 ('how to use'), 食べ方 ('way of eating'), 過ごし方 ('how to spend [time]') all follow the same pattern. The frame [noun]をする with abstract or manner-type nouns is how Japanese says 'have a [noun]' or 'put on a [noun]' — like 顔をする ('make a face'), スタイルをしている ('have a figure'), 服装をする ('be dressed in'). 個性的 ('distinctive, individualistic') is a na-adjective; the ~的 suffix is a Sino-Japanese affix that turns nouns into adjective-like modifiers, similar in feel to English '-ish' or '-ic.'